2.1 An auxiliary automatic sequence

We start by defining a substitution $\varphi $ on three letters:

(2-1) $$ \begin{align} \begin{array}{llll} \varphi:&{\mathtt a}\mapsto {\mathtt a}{\mathtt b}{\mathtt c},&{\mathtt b}\mapsto {\mathtt a}{\mathtt c},& {\mathtt c}\mapsto{\mathtt b}. \end{array} \end{align} $$

The morphism $\varphi $ can be extended to $\{{\mathtt a},{\mathtt b},{\mathtt c}\}^{\mathbb N}$ by concatenation, and we denote this extension by $\varphi $ again. The unique fixed point (of length greater than $0$ ) of $\varphi $ is

$$ \begin{align*} {\mathbf A}={\mathtt a}{\mathtt b}{\mathtt c}{\mathtt a}{\mathtt c}{\mathtt b}{\mathtt a}{\mathtt b}{\mathtt c}{\mathtt b}{\mathtt a}{\mathtt c} {\mathtt a}{\mathtt b}{\mathtt c}{\mathtt a}{\mathtt c}{\mathtt b}{\mathtt a}{\mathtt c}{\mathtt a}{\mathtt b}{\mathtt c}{\mathtt b} {\mathtt a}{\mathtt b}{\mathtt c}{\mathtt a}{\mathtt c}{\mathtt b}{\mathtt a}{\mathtt b}{\mathtt c}{\mathtt b}{\mathtt a}{\mathtt c} {\mathtt a}{\mathtt b}{\mathtt c}{\mathtt b}{\mathtt a}{\mathtt b}{\mathtt c}{\mathtt a}{\mathtt c}{\mathtt b}{\mathtt a}{\mathtt c}\cdots. \end{align*} $$

This fixed point is a morphic, or substitutive, sequence [Reference Allouche and Shallit4, Ch. 7]. As a fixed point of $\varphi $ (without having to apply a coding of the fixed point) it is even pure morphic. The sequence ${\mathbf A}$ is in fact $2$ -automatic, which follows from Berstel [Reference Berstel7, Corollary 4]. It is a ‘hidden automatic sequence’, as treated very recently by Allouche et al. [Reference Allouche, Dekking and Queffélec1]. In fact, every automatic sequence can also be written as a coding of a fixed point of a nonuniform morphism [Reference Allouche, Shallit, Raigorodskii and Rassias6] and in this sense is a ‘hidden’ automatic sequence. We restate a corresponding $2$ -uniform substitution found by Berstel in due course. The sequence ${\mathbf A}$ , called the ternary Thue–Morse sequence (for example, in the On-Line Encyclopedia of Integer Sequences (OEIS) [Reference Sloane26, A036577]), Istrail squarefree sequence [Reference Allouche, Dekking and Queffélec1, Reference Istrail19], or vtm [Reference Blanchet-Sadri, Currie, Rampersad and Fox10], is well known. Citing Dekking [Reference Dekking14], we note that it appears in fact 12 times on the OEIS [Reference Sloane26], featuring all renamings of the letters corresponding to permutations of the sets $\{0,1,2\}$ and $\{1,2,3\}$ . These 12 entries are A005679, A007413, and A036577–A036586. The sequence ${\mathbf A}$ encodes the gaps between consecutive ${\mathtt 1}$ s in ${\mathbf t}$ [Reference Blanchet-Sadri, Currie, Rampersad and Fox10]. Thue [Reference Thue27] showed that ${\mathbf A}$ is squarefree, while Rao et al. [Reference Rao, Rigo and Salimov24] later proved the stronger statement that ${\mathbf A}$ even avoids $2$ -binomial squares [Reference Rigo and Salimov25], thus settling in particular the question whether $2$ -abelian squares are avoidable over a three-letter alphabet. We use the squarefreeness property in our proof of Theorem 1.1.

We have the following important relation between ${\mathbf A}$ and our problem.

Lemma 2.2. The Thue–Morse sequence ${\mathbf t}$ can be recovered from ${\mathbf A}$ via the substitution

(2-2) $$ \begin{align} f:{\mathtt a}\mapsto {\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}{\mathtt 1}{\mathtt 0},\quad {\mathtt b}\mapsto {\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0},\quad {\mathtt c}\mapsto {\mathtt 0}{\mathtt 1}, \end{align} $$

by concatenation: we have

(2-3) $$ \begin{align} {\mathbf t}=f({\mathbf A}_0)f({\mathbf A}_1)\cdots. \end{align} $$

We prove this in a moment. From this observation, noting also that each of the three words $f({\mathtt a})$ , $f({\mathtt b})$ , and $f({\mathtt c})$ begins with ${\mathtt 0}{\mathtt 1}$ , we see that we can extract from ${\mathbf A}$ the sequence of gaps between occurrences of the factor ${\mathtt 0}{\mathtt 1}$ in $\mathbf t$ : each ${\mathtt a}$ yields two consecutive gaps of size $3$ , each ${\mathtt b}$ yields a gap of size $4$ , and each ${\mathtt c}$ a gap of size $2$ .

Proof of Lemma 2.2

We prove that for $k\geq 1$ ,

(2-4) $$ \begin{align} {\mathbf t}_{[0,6\cdot 2^k]}= f({\mathbf A}_0)f({\mathbf A}_1)\cdots f({\mathbf A}_{L_k}), \end{align} $$

where $L_k=3\cdot 2^{k-1}-1$ , by induction. The case $k=1$ is just the trivial identity ${\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}{\mathtt 1}{\mathtt 0}{\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}{\mathtt 0}{\mathtt 1}=f({\mathtt a})f({\mathtt b})f({\mathtt c})$ .

Note that $\tau (f({\mathtt a}))=f({\mathtt a})f({\mathtt b})f({\mathtt c})$ , $\tau (f({\mathtt b}))=f({\mathtt a})f({\mathtt c})$ , and $\tau (f({\mathtt c}))=f({\mathtt b})$ . Extending f by concatenation, for convenience of notation, to words over $\{{\mathtt a},{\mathtt b},{\mathtt c}\}$ , we obtain $\tau (f(x))=f(\varphi (x))$ for each $x\in \{{\mathtt a},{\mathtt b},{\mathtt c}\}$ . An application of the morphism $\tau $ to both sides of (2-4) yields

$$ \begin{align*} \begin{aligned} {\mathbf t}_{[0,6\cdot 2^{k+1})} &=\tau(f({\mathbf A}_0))\cdots\tau(f({\mathbf A}_{L_k})) =f(\varphi({\mathbf A}_0))\cdots f(\varphi({\mathbf A}_{L_k})) \\&=f(\varphi({\mathbf A}_0\cdots{\mathbf A}_{L_k})) =f({\mathbf A}_1\cdots{\mathbf A}_{L_{k+1}}) =f({\mathbf A}_0)\cdots f({\mathbf A}_{L_{k+1}}). \end{aligned} \end{align*} $$

Note that we see by induction that $\varphi ^k({\mathtt a}{\mathtt b}{\mathtt c})$ contains each of the three letters $2^k$ times, hence the numbers $L_k$ . This proves the lemma.

Since each of the words $f({\mathtt a})$ , $f({\mathtt b})$ , and $f({\mathtt c})$ starts with ${\mathtt 0}{\mathtt 1}$ , the differences $a_j=k_{j+1}-k_j$ between successive occurrences of ${\mathtt 0}{\mathtt 1}$ in $\mathbf t$ are easily obtained from ${\mathbf A}$ by the substitution

(2-5) $$ \begin{align} r:{\mathtt a}\mapsto {\mathtt 3}{\mathtt 3},\quad{\mathtt b}\mapsto {\mathtt 4},\quad{\mathtt c}\mapsto {\mathtt 2}. \end{align} $$

Here each ${\mathtt a}$ yields two blocks ${\mathtt 0}{\mathtt 1}$ , and each ${\mathtt b}$ or ${\mathtt c}$ one block.

Let ${\mathbf B}$ be the sequence of gaps between consecutive occurrences of ${\mathtt 0}{\mathtt 1}$ in the Thue–Morse sequence, and $\check {{\mathbf B}}$ the corresponding sequence for ${\mathtt 1}{\mathtt 0}$ .

Lemma 2.3. The sequence ${\mathbf B}$ is a morphic sequence, given by the substitution $\psi $ on the four letters ${\mathtt a},{\bar {\mathtt a}},{\mathtt b},{\mathtt c}$ , together with the coding p, given by

(2-6) $$ \begin{align} \begin{array}{lllll} \psi:&{\mathtt a}\mapsto{\mathtt a}{\bar{\mathtt a}},&{\bar{\mathtt a}}\mapsto{\mathtt b}{\mathtt c},&{\mathtt b}\mapsto{\mathtt a}{\bar{\mathtt a}}{\mathtt c},&{\mathtt c}\mapsto{\mathtt b},\\ p:&{\mathtt a}\mapsto{\mathtt 3},&{\bar{\mathtt a}}\mapsto{\mathtt 3},&{\mathtt b}\mapsto{\mathtt 4},&{\mathtt c}\mapsto{\mathtt 2}. \end{array} \end{align} $$

Let ${\bar {\mathbf B}}$ denote the fixed point of $\psi $ starting with ${\mathtt a}$ .

The sequence $\check {\mathbf B}$ is morphic. More precisely, it is the image of ${\bar {\mathbf B}}$ under the morphism

$$ \begin{align*} \check p:{\mathtt a}\mapsto {\mathtt 2}{\mathtt 4},\quad{\bar{\mathtt a}}\mapsto {\mathtt 3}{\mathtt 3},\quad {\mathtt b}\mapsto {\mathtt 2}{\mathtt 3}{\mathtt 3},\quad {\mathtt c}\mapsto {\mathtt 4}. \end{align*} $$

Note that ${\bar {\mathbf B}}$ is the pointwise limit of the finite words $\psi ^k({\mathtt a})$ , and begins as follows:

$$ \begin{align*} {\bar{\mathbf B}}={\mathtt a}{\bar{\mathtt a}}{\mathtt b}{\mathtt c}{\mathtt a}{\bar{\mathtt a}}{\mathtt c}{\mathtt b}{\mathtt a}{\bar{\mathtt a}}{\mathtt b}{\mathtt c}{\mathtt b}{\mathtt a}{\bar{\mathtt a}}{\mathtt c} {\mathtt a}{\bar{\mathtt a}}{\mathtt b}{\mathtt c}{\mathtt a}{\bar{\mathtt a}}{\mathtt c}{\mathtt b}{\mathtt a}{\bar{\mathtt a}}{\mathtt c}{\mathtt a}{\bar{\mathtt a}}{\mathtt b}{\mathtt c}{\mathtt b} {\mathtt a}{\bar{\mathtt a}}{\mathtt b}{\mathtt c}{\mathtt a}{\bar{\mathtt a}}{\mathtt c}{\mathtt b}{\mathtt a}{\bar{\mathtt a}}{\mathtt b}{\mathtt c}{\mathtt b}{\mathtt a}{\bar{\mathtt a}}{\mathtt c} \cdots. \end{align*} $$

Proof. Let q be the morphism that replaces ${\mathtt a}$ by ${\mathtt a}{\bar {\mathtt a}}$ and leaves ${\mathtt b}$ and ${\mathtt c}$ unchanged. We show by induction on the length of a word C over ${\mathtt a}{\mathtt b}{\mathtt c}$ that

(2-7) $$ \begin{align} q(\varphi(C))=\psi(q(C)). \end{align} $$

This is clear for words of length $1$ , since $q(\varphi ({\mathtt a}))={\mathtt a}{\bar {\mathtt a}}{\mathtt b}{\mathtt c}=\psi (q({\mathtt a}))$ , $q(\varphi ({\mathtt b}))={\mathtt a}{\bar {\mathtt a}}{\mathtt c}=\psi (q({\mathtt b}))$ , and $q(\varphi ({\mathtt c}))={\mathtt b}=\psi (q({\mathtt c}))$ . Appending a letter $x\in \{{\mathtt a},{\mathtt b},{\mathtt c}\}$ to a word C for which the identity (2-7) already holds, we obtain

$$ \begin{align*} \begin{aligned} q(\varphi(Cx))&=q(\varphi(C)\varphi(x))=q(\varphi(C))q(\varphi(x)) \\&=\psi(q(C))\psi(q(x))=\psi(q(C)q(x))=\psi(q(Cx)) \end{aligned} \end{align*} $$

and therefore (2-7) for C replaced by $Cx$ . Next, we prove by induction, using (2-7), that

$$ \begin{align*}q(\varphi^k({\mathtt a}))=\psi^k(q({\mathtt a})).\end{align*} $$

Clearly, this holds for $k=1$ . For $k\geq 2$ , we obtain

$$ \begin{align*}q(\varphi^k({\mathtt a}))= q(\varphi(\varphi^{k-1}({\mathtt a}))) =\psi(q(\varphi^{k-1}({\mathtt a}))) =\psi(\psi^{k-1}(q({\mathtt a}))) =\psi^k(q({\mathtt a})). \end{align*} $$

Noting that $q({\mathtt a})=\psi ({\mathtt a})$ and $p\circ q=r$ , the proof of the first part of Lemma 2.3 is complete.

We proceed to the second part, concerning $\check {\mathbf B}$ . Note that by Corollary 7.7.5 in [Reference Allouche and Shallit4] we only have to prove that $\check {\mathbf B}=\check p({\bar {\mathbf B}})$ .

Let

$$ \begin{align*} \check f: {\mathtt a}\mapsto {\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}{\mathtt 1}{\mathtt 0},\quad {\bar{\mathtt a}}\mapsto {\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}{\mathtt 0}{\mathtt 1},\quad {\mathtt b}\mapsto {\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}{\mathtt 1}{\mathtt 0}{\mathtt 0}{\mathtt 1},\quad {\mathtt c}\mapsto {\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}, \end{align*} $$

and extend this function to words (finite or infinite) over $\{{\mathtt a},{\bar {\mathtt a}},{\mathtt b},{\mathtt c}\}$ by concatenation.

Applying $\tau $ , we see by direct computation that

$$ \begin{align*}\tau(\check f({\mathtt a}))={\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}{\mathtt 1}{\mathtt 0}{\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}{\mathtt 0}{\mathtt 1} = \check f({\mathtt a})\check f({\bar{\mathtt a}}) =\check f(\psi({\mathtt a})), \end{align*} $$

and analogously, we get $\tau (\check f(x))=\check f(\psi (x))$ for each letter $x\in \{{\bar {\mathtt a}},{\mathtt b},{\mathtt c}\}$ . Applying this letter by letter, we obtain

$$ \begin{align*} \tau(\check f(w))=\check f(\psi(w)) \end{align*} $$

for every finite word over $\{{\mathtt a},{\bar {\mathtt a}},{\mathtt b},{\mathtt c}\}$ . By induction, we obtain

$$ \begin{align*}\tau^k(\check f({\mathtt a}))=\check f(\psi^k({\mathtt a})),\end{align*} $$

using the step

$$ \begin{align*}\tau^{k+1}(\check f({\mathtt a}))= \tau(\tau^k(\check f({\mathtt a}))) =\tau(\check f(\psi^k({\mathtt a}))) =\check f(\psi^{k+1}({\mathtt a})). \end{align*} $$

Noting that $\check f({\mathtt a})$ begins with ${\mathtt 0}$ , we obtain ${\mathbf t}=\check f({\bar {\mathbf B}})$ . In other words, the sequence ${\bar {\mathbf B}}$ yields the decomposition ${\mathbf t}=x_0x_1\cdots $ of the Thue–Morse sequence into return words of ${\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}$ , where $x_j=\check f({\bar {\mathbf B}}_j)$ . From this decomposition we can easily read off the sequence of gaps between occurrences of ${\mathtt 1}{\mathtt 0}$ , since this word appears in each of the four return words, and the first occurrence always takes place at the same position, which is $2$ . In this way, we obtain the gaps $2$ and $3$ from the return word $\check f({\mathtt a})$ each time ${\mathtt a}$ appears in ${\bar {\mathbf B}}$ . Analogously, ${\bar {\mathtt a}}$ yields the gaps $3$ and $3$ , the letter ${\mathtt b}$ the gaps $2$ , $3$ , and $3$ , and finally ${\mathtt c}$ yields the gap $4$ . This proves the second part of Lemma 2.3.

2.2 Factors of ${\mathbf B}$ appearing at positions in a residue class

The main step in our proof of Theorem 1.1 is given by the following proposition. For completeness, we let $\psi ^0$ denote the identity, so that $\psi ^0(w)=w$ for all words w over $\{{\mathtt a},{\bar {\mathtt a}},{\mathtt b},{\mathtt c}\}$ .

In the remainder of this section, we prove this proposition. We work with the fourth iteration ${\sigma }=\psi ^4$ of the substitution $\psi $ : we have

(2-8) $$ \begin{align} \begin{array}{l@{\hspace{0.4em}}l@{\hspace{1.5em}}l@{\hspace{0.4em}}l} {\sigma}({\mathtt a})&={\mathtt a}{\bar{\mathtt a}}{\mathtt b}{\mathtt c}{\mathtt a}{\bar{\mathtt a}}{\mathtt c}{\mathtt b}{\mathtt a}{\bar{\mathtt a}}{\mathtt b}{\mathtt c}{\mathtt b}{\mathtt a}{\bar{\mathtt a}}{\mathtt c},& {\sigma}({\bar{\mathtt a}})&={\mathtt a}{\bar{\mathtt a}}{\mathtt b}{\mathtt c}{\mathtt a}{\bar{\mathtt a}}{\mathtt c}{\mathtt b}{\mathtt a}{\bar{\mathtt a}}{\mathtt c}{\mathtt a}{\bar{\mathtt a}}{\mathtt b}{\mathtt c}{\mathtt b},\\ {\sigma}({\mathtt b})&={\mathtt a}{\bar{\mathtt a}}{\mathtt b}{\mathtt c}{\mathtt a}{\bar{\mathtt a}}{\mathtt c}{\mathtt b}{\mathtt a}{\bar{\mathtt a}}{\mathtt b}{\mathtt c}{\mathtt b}{\mathtt a}{\bar{\mathtt a}}{\mathtt c}{\mathtt a}{\bar{\mathtt a}}{\mathtt b}{\mathtt c}{\mathtt b},& {\sigma}({\mathtt c})&={\mathtt a}{\bar{\mathtt a}}{\mathtt b}{\mathtt c}{\mathtt a}{\bar{\mathtt a}}{\mathtt c}{\mathtt b}{\mathtt a}{\bar{\mathtt a}}{\mathtt c}. \end{array} \end{align} $$

We have the following explicit formulas for the lengths of ${\sigma }^k(x)$ , where $x\in \{{\mathtt a},{\bar {\mathtt a}},{\mathtt b},{\mathtt c}\}$ :

(2-9) $$ \begin{align} \begin{aligned} a_k&:= \lvert{\sigma}^k({\mathtt a})\rvert=\lvert{\sigma}^k({\bar{\mathtt a}})\rvert=16^k,\\ b_k&:=\lvert{\sigma}^k({\mathtt b})\rvert=\frac{4\cdot16^k-1}3,\\ c_k&:=\lvert{\sigma}^k({\mathtt c})\rvert=\frac{2\cdot16^k+1}3. \end{aligned} \end{align} $$

The proof of this identity is based on the formula

$$ \begin{align*} \left(\begin{matrix}4&4&4&4\\4&4&4&4\\5&5&6&5\\3&3&2&3\end{matrix}\right)^k \!\!=\left(\begin{matrix} 16^k/4&16^k/4&16^k/4&16^k/4\\ 16^k/4&16^k/4&16^k/4&16^k/4\\ (16^k-1)/3&(16^k-1)/3&(16^k+2)/3&(16^k-1)/3\\ (16^k+2)/6&(16^k+2)/6&(16^k-4)/6&(16^k+2)/6 \end{matrix}\right)\!, \end{align*} $$

valid for $k\geq 1$ , which takes care of the numbers of the letters ${\mathtt a}$ , ${\bar {\mathtt a}}$ , ${\mathtt b}$ , and ${\mathtt c}$ in ${\sigma }^k({\mathtt a})$ , ${\sigma }^k({\bar {\mathtt a}})$ , ${\sigma }^k({\mathtt b})$ , and ${\sigma }^k({\mathtt c})$ . This formula can be proved easily by induction. Moreover, (2-9) also holds for $k=0$ .

By applying ${\sigma }^k$ on the first line of (2-8), we see that each letter in $\{{\mathtt a},{\bar {\mathtt a}},{\mathtt b},{\mathtt c}\}$ is replaced by a word having the respective lengths $a_k$ , $a_k$ , $b_k$ , $c_k$ . For each $\nu \geq 0$ , it follows that the factor $\sigma ^{\nu }({\mathtt a})$ , of length $a_{\nu }=16^{\nu }$ , can be found at the following positions in ${\sigma }^{\nu +1}({\mathtt a})$ :

(2-10) $$ \begin{align} \begin{array}{ll} A^{(\nu,0)}:= 0,& A^{(\nu,1)}:= 4\cdot16^{\nu},\\ A^{(\nu,2)}:=8\cdot16^{\nu},& \displaystyle A^{(\nu,3)}:=12\cdot16^{\nu}+\frac{4\cdot16^{\nu}-1}3. \end{array} \end{align} $$

We may repeat this for $\nu -1,\nu -2,\ldots ,\mu $ , where $\mu \le \nu $ is a given natural number, from which we obtain the following statement. For all integers $0\leq \mu \leq \nu $ and all $\boldsymbol \varepsilon =(\varepsilon _{\mu },\varepsilon _{\mu +1},\ldots ,\varepsilon _{\nu }) \in \{0,1,2,3\}^{\nu -\mu +1}$ , the factor $\sigma ^{\mu }({\mathtt a})$ of length $16^{\mu }$ can be found at the position

(2-11) $$ \begin{align} N_{\boldsymbol\varepsilon}:= A^{(\mu,\varepsilon_{\mu})}+A^{(\mu+1,\varepsilon_{\mu+1})}+\cdots+A^{(\nu,\varepsilon_{\nu})} \end{align} $$

in ${\bar {\mathbf B}}$ . There are other positions where the factor $\sigma ^{\mu }({\mathtt a})$ appears, but for our proof it is sufficient to consider these special positions. We show that we can find one among these indices $N_{\boldsymbol \varepsilon }$ in a given residue class $a+m\mathbb Z$ .

Let us sketch the remainder of the proof. The case where m is even causes mild difficulties. We therefore write $m=2^kd$ , where d is odd, and proceed in two steps. As a first step, we find integers $\mu $ , $\nu $ , and $\varepsilon _{\mu },\varepsilon _{\mu +1},\ldots ,\varepsilon _{\lambda -1}\in \{0,1,2,3\}$ , such that $N_{\varepsilon _{\mu },\varepsilon _{\mu +1},\ldots ,\varepsilon _{\lambda -1}}$ lies in any given residue class modulo $2^k$ . The second step involves refining the description by appending a sequence $(\varepsilon _{\lambda },\ldots ,\varepsilon _{\nu -1})\in \{0,1,2\}^{\nu -\lambda }$ . Since we exclude the digit $\varepsilon _i=3$ , and we take care that $16^{\mu }\geq 2^k$ , we have

$$ \begin{align*}N_{\varepsilon_{\mu},\ldots,\varepsilon_{\nu-1}}\equiv N_{\varepsilon_{\mu},\ldots,\varepsilon_{\lambda-1}}\bmod 2^k.\end{align*} $$

We choose the integers $\varepsilon _j$ for $\lambda \leq j<\mu $ in such a way that any given residue class modulo d (note that d is odd) is hit. Due to the excluded digit $3$ , this is a missing digit problem, and a short argument including exponential sums finishes this step. Combining these two steps, we see that every residue class modulo $2^kd$ is reached. We now go into the details.

First step: hitting a residue class modulo $2^k$

We are interested in appearances of the initial segment $\sigma ^{\mu }({\mathtt a})$ in ${\bar {\mathbf B}}$ at positions lying in the residue class $a+2^k\mathbb Z$ . Let us assume in the following that

(2-12) $$ \begin{align} 16^{\mu}\geq 2^k. \end{align} $$

This lower bound on $\mu $ does not cause any problems.

We choose $\lambda>\mu $ in a moment, and we set $\varepsilon _{\mu }=\cdots =\varepsilon _{\lambda -1}=3$ . Let us consider the integers $\alpha _0:=0$ , and for $1\leq \ell \leq \lambda -\mu $ ,

$$ \begin{align*}\alpha_{\ell}:= N_{\varepsilon_{\mu},\ldots,\varepsilon_{\mu+\ell-1}}.\end{align*} $$

Assume that $0\leq \ell <\lambda -\mu $ . By (2-11) and (2-12), we have

$$ \begin{align*} \begin{aligned} \alpha_{\ell+1}-\alpha_{\ell} &=12\cdot 16^{\mu+\ell}+\frac{4\cdot 16^{\mu+\ell}-1}3 \equiv \frac{4\cdot 16^{\mu+\ell}-1}3\bmod 2^{\ell} \\[4pt]& \equiv \sum_{0\leq j\leq 2\mu+2\ell}4^j \bmod 2^{\ell} \equiv \sum_{0\leq j<2\mu}4^j \bmod 2^{\ell}. \end{aligned} \end{align*} $$

The latter sum is an odd integer, and independent of $\ell $ . It follows that $(\alpha _{\ell })_{0\leq \ell \leq \lambda -\mu }$ is an arithmetic progression modulo $2^k$ , where the common difference is odd; choosing $\lambda \geq \mu +2^k$ , we see that $(\alpha _{\ell })_{0\leq \ell <\lambda -\mu }$ hits every residue class modulo $2^k$ . We summarize the first step in the following lemma.

Second step: a discrete Cantor set–missing digits

We follow the paper [Reference Erdős, Mauduit and Sárközy17] by Erdős et al., who studied integers with missing digits in residue classes. Let ${\mathcal W}_{\lambda }$ be the set of nonnegative multiples of $16^{\lambda }$ having only the digits $0,4$ , and $8$ in their base- $16$ expansion. Set

$$ \begin{align*}U(\alpha)=\frac 13\sum_{0\leq k\leq 2}\operatorname{{\mathrm{e}}}(4k\alpha)\quad\mbox{and}\quad G(\alpha,\lambda,\nu)=\frac1{3^{\nu-\lambda}}\sum_{\substack{0\leq j<16^{\nu}\\ j\in {\mathcal W}}} \operatorname{{\mathrm{e}}}({\kern1.5pt}j\alpha), \end{align*} $$

where $\operatorname {{\mathrm {e}}}(x)=\exp (2\pi i x)$ . Note that the elements $j\in {\mathcal W}_{\lambda }$ have the form $j=\sum _{\lambda \leq k<\eta }4\,\varepsilon _k16^k$ , where $\eta \geq 0$ and $\varepsilon _k\in \{0,1,2\}$ for $\lambda \leq k<\eta $ . In particular, ${\mathcal W}_{\lambda }\cap [0,16^{\eta })$ has $3^{\eta -\lambda }$ elements for $\eta \geq \lambda $ . We obtain

(2-13) $$ \begin{align} G(\alpha,\lambda,\nu)&= \frac 1{3^{\nu-\lambda}} \hspace{-1em} \sum_{(\varepsilon_{\lambda},\ldots,\varepsilon_{\nu-1})\in\{1,2,3\}^{\nu-\lambda}} \hspace{-1em} \operatorname{{\mathrm{e}}}(4\,\varepsilon_{\lambda} 16^{\lambda}\alpha+\cdots+4\,\varepsilon_{\nu-1}16^{\nu-1}\alpha) \nonumber\\&= \prod_{\lambda\leq r<\nu} \tfrac 13(\operatorname{{\mathrm{e}}}(0\cdot 16^r\alpha)+\operatorname{{\mathrm{e}}}(4\cdot 16^r\alpha)+\operatorname{{\mathrm{e}}}(8\cdot 16^r\alpha)) \nonumber\\&= \prod_{\lambda\leq r<\nu} U(16^r\alpha). \end{align} $$

The purpose of this section is to prove the following lemma.

In order to prove this, we first show that it is sufficient to prove the following auxiliary result (compare [Reference Erdős, Mauduit and Sárközy17, formula (4.3)]).

Lemma 2.8. Assume that $d\geq 1$ is an odd integer, and $\ell \in \{1,\ldots ,d-1\}$ . Let $\lambda \geq 0$ be an integer. Then

$$ \begin{align*}\lim_{\nu\rightarrow \infty}G\bigg(\frac \ell d,\lambda,\nu\bigg)=0.\end{align*} $$

In fact, by the orthogonality relation

$$ \begin{align*} \frac 1d\sum_{0\leq n<d}\operatorname{{\mathrm{e}}}(nk/d)=\left\{\!\begin{array}{l} \!1\quad\mbox{if }d\mid k,\\ \!0\quad\mbox{otherwise}, \end{array}\right. \end{align*} $$

we have

(2-14) $$ \begin{align} & \frac1{3^{\nu-\lambda}} \#\{ 0\leq j<16^{\nu}: j\in {\mathcal W}, j\equiv a\bmod d \} -\frac 1d \nonumber\\&\quad= \frac 1d \frac1{3^{\nu-\lambda}} \sum_{0\leq \ell<d} \sum_{\substack{0\leq j<16^{\nu}\\ j\in{\mathcal W}}} \operatorname{{\mathrm{e}}}(\ell({\kern1.5pt}j-a)/d) -\frac 1d\nonumber\\ &\quad= \frac 1d \sum_{1\leq \ell<d} \operatorname{{\mathrm{e}}}(-\ell a/d) \frac1{3^{\nu-\lambda}} \sum_{\substack{0\leq j<16^{\nu}\\ j\in{\mathcal W}}} \operatorname{{\mathrm{e}}}({\kern1.5pt}j\ell/d) \leq \sum_{1\leq \ell<d} \bigg\lvert G\bigg(\frac \ell d,\lambda,\nu\bigg) \bigg\rvert. \end{align} $$

If $G(\ell /d,\lambda ,\nu )$ converges to zero as $\nu \rightarrow \infty $ , for all $\ell \in \{1,\ldots ,d-1\}$ , the last sum in (2-14) is eventually smaller than $1/d$ . Consequently, the number of $j\in \{0,\ldots,16^\nu-1\}$ such that $j\in\mathcal W_\lambda$ and $j\equiv a\bmod d$ diverges to $\infty$ as $\nu$ approaches $\infty$ .

Proof of Lemma 2.8

By (2-14), we have to show that the product

(2-15) $$ \begin{align} \prod_{\lambda\leq r<\nu}U(16^r\ell/d) = \prod_{\lambda\leq r<\nu}(1+\operatorname{{\mathrm{e}}}(4\cdot 16^r\ell/d)+\operatorname{{\mathrm{e}}}(8\cdot 16^r\ell/d)) \end{align} $$

converges to zero as $\nu \rightarrow \infty $ . To this end, we use the following lemma [Reference Delange15] by Delange.

Lemma 2.9 (Delange)

Assume that $q\geq 2$ is an integer and $z_1,\ldots ,z_{q-1}$ are complex numbers such that $\lvert z_j\rvert \leq 1$ for $1\leq j<q$ . Then

$$ \begin{align*}\bigg\lvert \frac1q(1+z_1+\cdots+z_{q-1})\bigg\rvert\leq 1-\frac1{2q}\max_{1\leq j<q}(1-{\operatorname{{\mathrm{Re}}}}\ z_j).\end{align*} $$

Since d is odd and $1\leq \ell <d$ , the integer $4k16^r\ell $ is not a multiple of d for $k\in \{1,2\}$ . It follows that $\operatorname {{\mathrm {Re}}} \operatorname {{\mathrm {e}}}(4k16^r\ell /d) \leq 1-\tilde \varepsilon $ for some $\tilde \varepsilon>0$ only depending on d.

Therefore each factor in (2-13) is smaller than $1-\varepsilon $ , where $\varepsilon>0$ does not depend on r. Consequently, by Lemma 2.9 the product (2-15) converges to zero. Lemma 2.8, and therefore Lemma 2.7, is proved.

Now we combine the two steps, corresponding to the cases (i) $2^k$ and (ii) d odd.

Let $k\geq 0$ and $d\geq 1$ be integers, and d odd. We are interested in a residue class $a+2^kd\,\mathbb Z$ , where $a\in \mathbb Z$ . Choose

$$ \begin{align*}a^{(1)}:= a\bmod 2^k\in\{0,\ldots,2^k-1\}.\end{align*} $$

Choose $\mu $ large enough such that $16^{\mu }\geq 2^k$ . By Lemma 2.6 there exists $\lambda \geq \mu $ such that

$$ \begin{align*}\kappa^{(1)}\equiv a^{(1)}\bmod 2^k,\end{align*} $$

where $\kappa ^{(1)}:= N_{\varepsilon _{\mu },\ldots ,\varepsilon _{\lambda -1}}$ and $\varepsilon _{\ell }=3$ for $\mu \leq \ell <\lambda $ . Next, choose

$$ \begin{align*}a^{(2)}:= (a-\kappa^{(1)})\bmod d.\end{align*} $$

By Lemma 2.7, the set ${\mathcal W}_{\lambda }\cap (a^{(2)}+d\,\mathbb Z)$ is not empty. Let $\sum _{\lambda \leq \ell <\nu }4\,\varepsilon _{\ell }16^{\ell }$ be an element, where $\varepsilon _{\ell }\in \{0,1,2\}$ for $\lambda \leq \ell <\nu $ . By (2-11) we have

$$ \begin{align*} \kappa:= N_{\varepsilon_{\mu},\ldots,\varepsilon_{\lambda-1},\varepsilon_{\lambda},\ldots,\varepsilon_{\nu-1}} &= \kappa^{(1)}+\kappa^{(2)}, \end{align*} $$

where

$$ \begin{align*} \kappa^{(2)}:= N_{\varepsilon_{\lambda},\ldots,\varepsilon_{\nu-1}}. \end{align*} $$

The integer $\kappa ^{(1)}$ lies in the residue class $a^{(1)}+2^k\mathbb Z$ by construction, while $\kappa ^{(2)}$ is divisible by $2^k$ , as no digit among $\varepsilon _{\lambda },\ldots ,\varepsilon _{\nu -1}$ equals $3$ . It follows that $\kappa \in a^{(1)}+2^k\mathbb Z=a+2^k\mathbb Z$ . Moreover, by (2-10),

$$ \begin{align*}\kappa^{(2)}=\sum_{\lambda\leq \ell<\nu}4\,\varepsilon_{\ell}16^{\ell}\in a^{(2)}+d\,\mathbb Z,\end{align*} $$

hence $\kappa =\kappa ^{(1)}+\kappa ^{(2)} \equiv \kappa ^{(1)}+(a-\kappa ^{(1)})\equiv a\bmod d$ .

Summarizing, we have $\kappa \in (a+2^k\mathbb Z)\cap (a+d\,\mathbb Z)$ . Since $2^k$ and d are coprime, which implies $2^k\mathbb Z\cap d\,\mathbb Z=2^kd\,\mathbb Z$ , we have $(a+2^k\mathbb Z)\cap (a+d\,\mathbb Z)=a+2^kd\,\mathbb Z$ (applying a shift by a) and therefore $\kappa \in a+2^kd\,\mathbb Z$ . This finishes the proof of Proposition 2.5.

2.3 Nonautomaticity of $\mathbf B$

In order to prove that $\mathbf B$ is not automatic, we use the characterization by the k-kernel: a sequence $(a_n)_{n\geq 0}$ is k-automatic if and only if the set

(2-16) $$ \begin{align} \{(a_{\ell+k^jn})_{n\geq 0}: j\geq 0, 0\leq \ell<k^j\} \end{align} $$

is finite.

We are now in a position to prove that any two arithmetic subsequences of ${\mathbf B}$ with the same modulus m and different shifts $\ell _1$ , $\ell _2$ are different: the sequences $({\mathbf B}(\ell _1+nm))_{n\geq 0}$ and $({\mathbf B}(\ell _2+nm))_{n\geq 0}$ cannot be equal. This proves, in particular, that the k-kernel is infinite and thus nonautomaticity of the gap sequence for ${\mathtt 0}{\mathtt 1}$ .

Let us assume, in order to obtain a contradiction, that the sequence ${\mathbf B}$ contains two identical arithmetic subsequences with common differences equal to m, indexed by $n\mapsto \ell _1+nm$ and $n\mapsto \ell _2+nm$ respectively, where $\ell _1<\ell _2$ . Let $r=\ell _2-\ell _1$ , and choose $\mu $ large enough such that $16^{\mu }\geq 2r$ . By Proposition 2.5, the block $\sigma ^{\mu }({\mathtt a})$ appears in ${\bar {\mathbf B}}$ at positions that hit each residue class. In particular, for each $s\in \{0,\ldots ,r-1\}$ , we choose the residue class $\ell _1-s+m\mathbb Z$ , and we can find an index n such that $\sigma ^{\mu }({\mathtt a})$ appears at position $\ell _1-s+nm$ in ${\bar {\mathbf B}}$ . Since $16^m\geq 2r>s$ , this means that $\ell _1+mn$ hits the $s\, $ th letter in $\sigma ^{\mu }({\mathtt a})$ , or in symbols,

$$ \begin{align*}{\bar{\mathbf B}}_{\ell_1+nm}=\sigma^{\mu}({\mathtt a})\rvert_s.\end{align*} $$

Since $s+r$ is still in the range $[0,16^{\mu })$ , we also have

$$ \begin{align*}{\bar{\mathbf B}}_{\ell_2+nm}=\sigma^{\mu}({\mathtt a})\rvert_{s+r}\end{align*} $$

for the same index n. Applying the coding p defined in (2-6), and our equality assumption, we see that

$$ \begin{align*} {\mathbf B}_s= p(\sigma^{\mu}({\mathtt a})\rvert_s)= {\mathbf B}_{\ell_1+nm}={\mathbf B}_{\ell_2+nm}= p(\sigma^{\mu}({\mathtt a})\rvert_{s+r})={\mathbf B}_{s+r}.\end{align*} $$

Carrying this out for all $s\in \{0,\ldots ,r-1\}$ , we see that the first $2r$ terms of ${\mathbf B}$ form a square. Now there are two cases to consider.

The case $r=1$ . Assume that ${\mathbf B}_{\ell _1+nm}={\mathbf B}_{\ell _1+1+nm}$ for all $n\geq 0$ . By Proposition 2.5, the positions where the prefix $\mathtt {3342}={\mathbf B}_0{\mathbf B}_1{\mathbf B}_2{\mathbf B}_3$ appears as a factor in ${\mathbf B}$ hit every residue class. In particular, there is an index n such that the block $\mathtt {3342}$ can be found at position $\ell _1-1+nm$ in ${\mathbf B}$ . This implies $\mathtt {3}={\mathbf B}_1={\mathbf B}_{\ell _1+nm}={\mathbf B}_{\ell _1+1+nm}={\mathbf B}_2=\mathtt {4}$ , a contradiction.

The case $r\ge 2$ . In this case we resort to the fact, proved below, that ${\mathbf B}$ does not contain squares of length greater than $2$ . Therefore we get a contradiction also in this case. In order to complete the proof that ${\mathbf B}$ is not automatic, it remains to prove (the second part of) the following result.

Summarizing, arithmetic subsequences of ${\mathbf B}$ with common difference m are distinct as soon as their offsets differ. In particular, for each integer $k\ge 2$ , the k-kernel of ${\mathbf B}$ is infinite. Therefore ${\mathbf B}$ is not automatic, which proves the case $w={\mathtt 0}{\mathtt 1}$ of Theorem 1.1.

2.4 Occurrences of general factors in ${\mathbf t}$

We begin with the case $w={\mathtt 1}{\mathtt 0}$ . We work with the Thue–Morse morphism $\tau :{\mathtt 0}\mapsto {\mathtt 0}{\mathtt 1}$ , ${\mathtt 1}\mapsto {\mathtt 1}{\mathtt 0}$ , defined in (1-1). First of all, we recall the well-known fact that $a_{k+1}=\tau ^{k+1}({\mathtt 0})$ can be constructed from $a_k=\tau ^k({\mathtt 0})$ by concatenating $a_k$ and its Boolean complement $\overline {a_k}$ (which replaces each ${\mathtt 0}$ by ${\mathtt 1}$ and each ${\mathtt 1}$ by ${\mathtt 0}$ ). The proof of this little fact is by an easy induction. For $k=0$ we have $a_1={\mathtt 0}{\mathtt 1}={\mathtt 0}\overline {{\mathtt 0}}$ . The case $k\geq 1$ makes use of the identity $\tau (\overline w)=\overline {\tau (w)}$ , valid for each word w over $\{{\mathtt 0},{\mathtt 1}\}$ , which follows from the special structure of the morphism $\tau $ . Applying this identity and the induction hypothesis, we obtain

$$ \begin{align*} a_{k+1}&=\tau(\tau^k({\mathtt 0}))=\tau(a_{k-1}\overline{a_{k-1}}) \\&=\tau(a_{k-1})\tau(\overline{a_{k-1}}) =\tau(a_{k-1})\overline{\tau(a_{k-1})} =a_k\overline{a_k}. \end{align*} $$

Using this, we show that for even $k\geq 0$ , the word $\tau ^k({\mathtt 0})$ is a palindrome. The case ${k=0}$ is trivial. If $a_k=\tau ^k({\mathtt 0})$ is a palindrome, then $a_{k+2}=\tau (\tau (a_k))=\tau (a_k\overline {a_k})=a_k\overline {a_k}\overline {a_k}a_k$ is clearly a palindrome too, and the statement follows by induction. In particular, we see from the above that

(2-17) $$ \begin{align} \tau^k({\mathtt 0})=\tau^{k-1}({\mathtt 0})\tau^{k-1}({\mathtt 1}),\quad \tau^k({\mathtt 1})=\tau^{k-1}({\mathtt 1})\tau^{k-1}({\mathtt 0}), \quad\mbox{for all }k\geq 0. \end{align} $$

Note that, by applying $\tau ^k$ on ${\mathbf t}$ , every ${\mathtt 0}$ gets replaced by $\tau ^k({\mathtt 0})$ and every ${\mathtt 1}$ by $\tau ^k({\mathtt 1})$ , and the result is again ${\mathbf t}$ since it is a fixed point of $\tau $ . It follows that

(2-18) $$ \begin{align} \begin{array}{rl} \mbox{for all }k\geq 0,&\mbox{ we have } {\mathbf t}=A_{{\mathbf t}_0}A_{{\mathbf t}_1}A_{{\mathbf t}_2}\cdots,\\ &\mbox{ where } A_x=\tau^k(x)\mbox{ for }x\in\{{\mathtt 0},{\mathtt 1}\}. \end{array} \end{align} $$

Let $(r_j)_{j\geq 0}$ be the increasing sequence of indices where ${\mathtt 1}{\mathtt 0}$ occurs in ${\mathbf t}$ . For k even, let $J=J(k)$ be the number of occurrences of ${\mathtt 1}{\mathtt 0}$ with indices less than or equal to $2^k-2$ . Note that $r_{J-1}=2^k-2$ . We read the (palindromic) sequence $a_k$ , of length $2^k$ , backwards; it follows that $(2^k-1-r_{J-1-j})_{0\leq j<J}$ is the increasing sequence of indices pointing to the letter ${\mathtt 1}$ in an occurrence of ${\mathtt 0}{\mathtt 1}$ in $a_k$ . Therefore,

$$ \begin{align*}(2^k-2-r_{J-1-j})_{0\leq j<J}\end{align*} $$

is the increasing sequence of indices where ${\mathtt 0}{\mathtt 1}$ occurs in $a_k$ . Consequently, by the definition of ${\mathbf B}$ as the differences of these indices, we obtain ${\mathbf B}_j=-r_{J-1-({\kern1.5pt}j+1)}+r_{J-1-j}$ for $0\leq j<J-1$ , and thus

(2-19) $$ \begin{align} r_{j+1}-r_j={\mathbf B}_{J-2-j,}\quad\mbox{for }0\leq j\le J-2. \end{align} $$

We have to prove that the sequence

$$ \begin{align*} \check{{\mathbf B}}=(r_{j+1}-r_j)_{j\geq 0} \end{align*} $$

is not automatic. More generally, we prove that any two arithmetic subsequences

$$ \begin{align*}L^{(1)}=(\check{{\mathbf B}}(\ell_1+nd))_{n\geq 0},\quad L^{(2)}=(\check{{\mathbf B}}(\ell_2+nd))_{n\geq 0},\end{align*} $$

where $d\geq 1$ and $\ell _1\neq \ell _2$ , are different. In order to obtain a contradiction, let us assume that $L^{(1)}=L^{(2)}$ , and let $k\geq 0$ be even. By (2-19), we get arithmetic subsequences $M_1$ , $M_2$ of ${\mathbf B}$ with common difference d, different offsets $m_1(k),m_2(k)\in \{0,\ldots ,d-1\}$ and length equal to $J(k)-1$ , such that

$$ \begin{align*}{\mathbf B}_{m_1(k)+nd}=M^{(1)}_j=M^{(2)}_j={\mathbf B}_{m_2(k)+nd},\quad\mbox{for }0\leq n\le J(k)-2.\end{align*} $$

Note the important fact that the offsets $m_j(k)$ are bounded by d. Since there are only $d(d-1)/2$ pairs $(a,b)\in \{0,\ldots ,d-1\}^2$ with $a\neq b$ , it follows that there are two different offsets $0\le \overline {m_1},\overline {m_2}<d$ with the following property: there are arbitrarily long arithmetic subsequences of ${\mathbf B}$ with indices of the form $\overline {m_1}+nd$ and $\overline {m_2}+nd$ respectively, taking the same values. This is just the statement that the infinite sequences $({\mathbf B}_{\overline {m_1}+nd})_{n\geq 0}$ and $({\mathbf B}_{\overline {m_2}+nd})_{n\geq 0}$ are equal. In the course of proving that ${\mathbf B}$ is not automatic (which is the case $w={\mathtt 0}{\mathtt 1}$ of Theorem 1.1) we proved that this is impossible, and we obtain a contradiction. The sequence $\check {{\mathbf B}}$ is therefore not automatic either, which finishes the case $w={\mathtt 1}{\mathtt 0}$ .

We proceed to the case $w={\mathtt 0}{\mathtt 0}$ . Let $(a_i)_{i\geq 0}$ be the increasing sequence of indices j such that ${\mathbf t}_j{\mathbf t}_{j+1}={\mathtt 0}{\mathtt 0}$ . Assume that $i\geq 0$ , and set $j:= a_i$ . We have $j\equiv 1\bmod 2$ , since ${\mathbf t}_{2j'}=\overline {{\mathbf t}_{2j'+1}}$ for all $j'\geq 0$ (where the overline denotes the Boolean complement, ${\mathtt 0}\mapsto {\mathtt 1}$ , ${\mathtt 1}\mapsto {\mathtt 0}$ ). Equality ${\mathbf t}_j={\mathbf t}_{j+1}$ (as needed) can therefore only occur at odd indices j, and we choose $j'\ge 0$ such that $j=2j'+1$ . Necessarily, ${\mathbf t}_{j'}={\mathtt 1}$ and ${\mathbf t}_{j'+1}={\mathtt 0}$ , since the identities ${\mathbf t}_{2j'+1}=\overline {{\mathbf t}_{j'}}$ and ${\mathbf t}_{2j'+2}={\mathbf t}_{j'+1}$ would produce an output ${\mathbf t}_{2j'+1}{\mathbf t}_{2j'+2}\neq {\mathtt 0}{\mathtt 0}$ in the other case. On the other hand, ${\mathbf t}_{j'}{\mathbf t}_{j'+1}={\mathtt 1}{\mathtt 0}$ indeed implies ${\mathbf t}_{2j'+1}{\mathbf t}_{2j'+2}={\mathtt 0}{\mathtt 0}$ . Each occurrence of ${\mathtt 0}{\mathtt 0}$ in ${\mathbf t}$ , at position j, therefore corresponds in a bijective manner to an occurrence of ${\mathtt 1}{\mathtt 0}$ at position $({\kern1.5pt}j-1)/2$ (which is an integer). It follows that the corresponding gap sequence equals $2\check {\mathbf B}$ , which is not automatic by the already proved case $w={\mathtt 1}{\mathtt 0}$ .

In a completely analogous manner, we can reduce the case $w={\mathtt 1}{\mathtt 1}$ to the case ${\mathtt 0}{\mathtt 1}$ , and the gap sequence equals $2{\mathbf B}$ , which is not automatic either.

We now reduce the case of general factors w of ${\mathbf t}$ of length at least $3$ to these four cases.

Lemma 2.11. For $x,y\in \{{\mathtt 0},{\mathtt 1}\}$ , let $(a^{xy}_k)_{k\ge 0}$ be the increasing sequence of indices j such that ${\mathbf t}_j{\mathbf t}_{j+1}=xy$ . We have

(2-20) $$ \begin{align} \begin{aligned} &a^{{\mathtt 0}{\mathtt 1}}_0<a^{{\mathtt 1}{\mathtt 0}}_0<a^{{\mathtt 0}{\mathtt 1}}_1<a^{{\mathtt 1}{\mathtt 0}}_1<a^{{\mathtt 0}{\mathtt 1}}_2<a^{{\mathtt 1}{\mathtt 0}}_2< \cdots\quad\mbox{and}\\ &a^{{\mathtt 1}{\mathtt 1}}_0<a^{{\mathtt 0}{\mathtt 0}}_0<a^{{\mathtt 1}{\mathtt 1}}_1<a^{{\mathtt 0}{\mathtt 0}}_1<a^{{\mathtt 1}{\mathtt 1}}_2<a^{{\mathtt 0}{\mathtt 0}}_2<\cdots. \end{aligned} \end{align} $$

Let w be a factor of ${\mathbf t}$ , of length at least $3$ . Choose $k\geq 0$ minimal such that w is a factor of some $a^{xy}_k=\tau ^k(x)\tau ^k(y)$ , where $x,y\in \{{\mathtt 0},{\mathtt 1}\}$ . By minimality, w is not a factor of $\tau ^k({\mathtt 0})$ or $\tau ^k({\mathtt 1})$ , using (2-17). Consequently, w appears at most once in each $a^{xy}_k$ . Next, we need the fact that ${\mathbf t}$ is overlap-free [Reference Berstel8, Reference Brown, Rampersad, Shallit and Vasiga13, Reference Thue27], meaning that it does not contain a factor of the form $axaxa$ , where $a\in \{{\mathtt 0},{\mathtt 1}\}$ and $x\in \{{\mathtt 0},{\mathtt 1}\}^{\ast }$ . We derive from this property that w cannot occur simultaneously in both members of any of the pairs

(2-21) $$ \begin{align} (a^{{\mathtt 0}{\mathtt 0}}_k,a^{{\mathtt 0}{\mathtt 1}}_k),\quad(a^{{\mathtt 0}{\mathtt 0}}_k,a^{{\mathtt 1}{\mathtt 0}}_k),\quad (a^{{\mathtt 1}{\mathtt 1}}_k,a^{{\mathtt 0}{\mathtt 1}}_k),\quad(a^{{\mathtt 1}{\mathtt 1}}_k,a^{{\mathtt 1}{\mathtt 0}}_k). \end{align} $$

For example, assume that w is a factor of both $a^{{\mathtt 0}{\mathtt 0}}_k$ and $a^{{\mathtt 0}{\mathtt 1}}_k$ . By minimality, as we had before,

$$ \begin{align*} \tau^k({\mathtt 0})\tau^k({\mathtt 0})=AwB,\quad \tau^k({\mathtt 0})\tau^k({\mathtt 1})=A'wB', \end{align*} $$

where A and $A'$ are initial segments of $\tau ^k({\mathtt 0})$ , and B (respectively, $B'$ ) are final segments of $\tau ^k({\mathtt 0})$ (respectively, $\tau ^k({\mathtt 1})$ ), and all of these segments are proper subwords of the respective words. We have $A\neq A'$ , since otherwise $\tau ^k({\mathtt 0})=\tilde wB=\tilde wB'=\tau ^k({\mathtt 1})$ for some $\tilde w$ that is not the empty word. This contradicts the fact that $\tau ^k({\mathtt 0})=\overline {\tau ^k({\mathtt 1})}$ . Let us, without loss of generality, assume that $\lvert A\rvert <\lvert A'\rvert $ . The first $2^k$ letters of $Aw$ and $A'w$ are equal, or in symbols,

(2-22) $$ \begin{align} (Aw)\rvert_{[0,2^k)}=(A'w)\rvert_{[0,2^k)}. \end{align} $$

We can therefore choose $a\in \{{\mathtt 0},{\mathtt 1}\}$ and $w_1,w_2\in \{{\mathtt 0},{\mathtt 1}\}^{\ast }$ in such a way that ${aw_1w_2=w}$ and $Aaw_1=A'$ . Then trivially $Aw=Aaw_1w_2=A'w_2$ , and since $\lvert A\rvert <2^k$ , $\lvert A'\rvert <2^k$ , it follows from (2-22) that $w_2=aw_3$ for some $w_3\in \{{\mathtt 0},{\mathtt 1}\}^{\ast }$ . Finally, the factor $A'w$ of ${\mathbf t}$ can be written as $A'w=Aaw_1w=Aaw_1aw_1w_2=Aaw_1aw_1aw_3$ , which contradicts the overlap-freeness of ${\mathbf t}$ . The other three cases, corresponding to the second, third and fourth pairs in (2-21), are analogous. We have therefore shown that the set of $A\in \{a^{{\mathtt 0}{\mathtt 0}}_k,a^{{\mathtt 0}{\mathtt 1}}_k,a^{{\mathtt 1}{\mathtt 0}}_k,a^{{\mathtt 1}{\mathtt 1}}_k\}$ such that w is a factor of A is a subset of either $\{a^{{\mathtt 0}{\mathtt 1}}_k,a^{{\mathtt 1}{\mathtt 0}}_k\}$ or $\{a^{{\mathtt 0}{\mathtt 0}}_k,a^{{\mathtt 1}{\mathtt 1}}_k\}$ .

First case. Let w be a factor of $a^{{\mathtt 0}{\mathtt 1}}_k$ , or of $a^{{\mathtt 1}{\mathtt 0}}_k$ . Assume first that w is a factor of $a^{{\mathtt 0}{\mathtt 1}}_k$ , but not of $a^{{\mathtt 1}{\mathtt 0}}_k$ . In this case, we show that the gap sequence for w is given by the gap sequence for $a^{{\mathtt 0}{\mathtt 1}}_k$ : (i) each occurrence of $a^{{\mathtt 0}{\mathtt 1}}_k$ yields exactly one occurrence of w (involving a constant shift); (ii) by (2-18), every occurrence of w takes place within a block of the form $a^{xy}_k$ ; (iii) only the block $a^{{\mathtt 0}{\mathtt 1}}_k$ is eligible. We prove that $a^{{\mathtt 0}{\mathtt 1}}_k$ appears exactly at positions $2^kj$ in ${\mathbf t}$ , where ${\mathbf t}_j{\mathbf t}_{j+1}={\mathtt 0}{\mathtt 1}$ . The easy direction follows from (2-18): each occurrence of ${\mathtt 0}{\mathtt 1}$ yields an occurrence of $a^{{\mathtt 0}{\mathtt 1}}_k$ , where the index has to be multiplied by $2^k$ . On the other hand, it is sufficient to show that $a^{{\mathtt 0}{\mathtt 1}}_k$ can only appear in positions $2^kj$ . Given this, there is no admissible choice for $({\mathbf t}_j,{\mathbf t}_{j+1})$ different from $({\mathtt 0},{\mathtt 1})$ , by (2-18). Suppose that we already know this for some $k\ge 0$ (the case $k=0$ being trivial). Assume that

(2-23) $$ \begin{align} a^{{\mathtt 0}{\mathtt 1}}_{k+1}=\tau^k({\mathtt 0})\tau^k({\mathtt 1})\tau^k({\mathtt 1})\tau^k({\mathtt 0}) \mbox{ appears on some position }\ell. \end{align} $$

Since $\tau ^k({\mathtt 0})\tau ^k({\mathtt 1})=a^{{\mathtt 0}{\mathtt 1}}_k$ , we know by hypothesis that $\ell \equiv 0\bmod 2^k$ . Assume that the case $\ell \equiv 2^k\bmod 2^{k+1}$ occurs. We set $\ell =(2j+1)2^k$ for some $j\geq 0$ . Our assumption (2-23) implies $\tau ^k({\mathtt 1})=\tau ^k({\mathbf t}_{2j+2})=\tau ^k({\mathbf t}_{j+1})$ and therefore ${\mathbf t}_{j+1}={\mathtt 1}$ , which implies that $\tau ^{k+1}({\mathbf t}_{j+1})=\tau ^k({\mathtt 1})\tau ^k({\mathtt 0})$ appears in position $\ell +2^k=(2j+2)2^k$ in ${\mathbf t}$ . This is incompatible with (2-23). In particular, the gap sequence for w, which is identical to the gap sequence for $a^{{\mathtt 0}{\mathtt 1}}_k$ , is given by $2^k{\mathbf B}$ , and therefore not automatic. Switching the roles of ${\mathtt 0}$ and ${\mathtt 1}$ in this proof, we also obtain nonautomaticity for the case where w is a factor of $a_k^{{\mathtt 1}{\mathtt 0}}$ , but not of $a_k^{{\mathtt 0}{\mathtt 1}}$ , with the sequence $2^k\check {\mathbf B}$ as the corresponding gap sequence.

Let w be a factor of both $a^{{\mathtt 0}{\mathtt 1}}_k$ and $a^{{\mathtt 1}{\mathtt 0}}_k$ . In this case, each occurrence of w in ${\mathbf t}$ takes place within a subblock of ${\mathbf t}$ of one of these two forms. By Lemma 2.11, combined with the above argument that occurrences of $a^{{\mathtt 0}{\mathtt 1}}_k$ (respectively, $a^{{\mathtt 1}{\mathtt 0}}_k$ ) in ${\mathbf t}$ take place at indices obtained from occurrences of ${\mathtt 0}{\mathtt 1}$ (respectively, ${\mathtt 1}{\mathtt 0}$ ), multiplied by $2^k$ , these blocks occur alternatingly. Assuming, in order to obtain a contradiction, that the gap sequence $(g_j)_{j\geq 0}$ for w is automatic, we obtain a new automatic sequence $(g_{2j}+g_{2j+1})_{j\geq 0}$ as the sum of two automatic sequences (note that the characterization involving the $2$ -kernel (2-16) immediately implies that $(g_{2j+\varepsilon })_{j\ge 0}$ , for $\varepsilon \in \{0,1\}$ , is automatic). By the alternating property, this is the gap sequence for $a^{{\mathtt 0}{\mathtt 1}}_k$ , which is not automatic, as we have just seen. A contradiction!

Second case. Let w be a factor of $a^{{\mathtt 0}{\mathtt 0}}_k$ or of $a^{{\mathtt 1}{\mathtt 1}}_k$ . This case is largely analogous. Assume that w is not a factor of $a^{{\mathtt 1}{\mathtt 1}}_k$ . As in the case $a^{{\mathtt 0}{\mathtt 1}}_k$ , the gap sequence for w in this case is identical to the gap sequence for $a^{{\mathtt 0}{\mathtt 0}}_k$ , and we only have to show that this sequence is not automatic. We know already that the gap sequence for ${\mathtt 0}{\mathtt 0}$ is not automatic. Therefore it suffices to prove that $\tau ^k({\mathtt 0})\tau ^k({\mathtt 0})$ can only appear at positions in ${\mathbf t}$ divisible by $2^k$ . Suppose that we already know this for some $k\ge 0$ (the case $k=0$ again being trivial). Assume that

(2-24) $$ \begin{align} a^{{\mathtt 0}{\mathtt 0}}_{k+1}=\tau^k({\mathtt 0})\tau^k({\mathtt 1})\tau^k({\mathtt 0})\tau^k({\mathtt 1}) \mbox{ appears on some position }\ell. \end{align} $$

Since $\tau ^k({\mathtt 0})\tau ^k({\mathtt 1})=a^{{\mathtt 0}{\mathtt 1}}_k$ , we know by hypothesis that $\ell \equiv 0\bmod 2^k$ . Assume that the case $\ell \equiv 2^k\bmod 2^{k+1}$ occurs. We set $\ell =(2j+1)2^k$ for some $j\geq 0$ . Our assumption (2-24) implies $\tau ^k({\mathtt 1})=\tau ^k({\mathbf t}_{2j+4})=\tau ^k({\mathbf t}_{j+2})$ and therefore ${\mathbf t}_{j+2}={\mathtt 1}$ , which implies that $\tau ^{k+1}({\mathbf t}_{j+2})=\tau ^k({\mathtt 1})\tau ^k({\mathtt 0})$ appears in position $\ell +3\cdot 2^k=(2j+4)2^k$ in ${\mathbf t}$ . In position $\ell $ , we therefore see the factor

$$ \begin{align*}\tau^k({\mathtt 0})\tau^k({\mathtt 1})\tau^k({\mathtt 0})\tau^k({\mathtt 1})\tau^k({\mathtt 0}),\end{align*} $$

which contradicts the overlap-freeness of ${\mathbf t}$ .

Again, the case where w is a factor of $a^{{\mathtt 1}{\mathtt 1}}_k$ , but not of $a^{{\mathtt 0}{\mathtt 0}}_k$ , is analogous; the case that it is a factor of both words can be handled as in the case $\{a^{{\mathtt 0}{\mathtt 1}}_k,a^{{\mathtt 1}{\mathtt 0}}_k\}$ , this time with the help of the second chain of inequalities in (2-20).

Summarizing, we have shown nonautomaticity for all gap sequences for factors w of ${\mathbf t}$ of length at least $2$ .

In order to finish the proof of Theorem 1.1, we still have to prove that the gap sequence is morphic for the ‘mixed cases’. That is, assume that w is a factor of two words of the form $a^{xy}_k$ , where $x,y\in \{{\mathtt 0},{\mathtt 1}\}$ , and where k is chosen minimal such that w is a factor of at least one of $a^{{\mathtt 0}{\mathtt 0}}_k,a^{{\mathtt 0}{\mathtt 1}}_k,a^{{\mathtt 1}{\mathtt 0}}_k,a^{{\mathtt 1}{\mathtt 1}}_k$ . Let us begin with the case $\{a^{{\mathtt 0}{\mathtt 1}}_k,a^{{\mathtt 1}{\mathtt 0}}_k\}$ . The positions where w appears in ${\mathbf t}$ are given by $2^kj+\sigma _0$ , where ${\mathtt b}_j{\mathtt b}_{j+1}={\mathtt 0}{\mathtt 1}$ , and $2^kj+\sigma _1$ , where ${\mathtt b}_j{\mathtt b}_{j+1}={\mathtt 1}{\mathtt 0}$ . Here $\sigma _0,\sigma _1$ are the positions where the word w appears in $a^{{\mathtt 0}{\mathtt 1}}_k$ and $a^{{\mathtt 1}{\mathtt 0}}_k$ , respectively. As before, this follows since ${\mathbf t}_j{\mathbf t}_{j+1}={\mathtt 0}{\mathtt 1}$ is equivalent to $({\mathbf t}_{\ell },\ldots ,{\mathbf t}_{\ell +2^{k+1}-1})=a^{{\mathtt 0}{\mathtt 1}}_k$ , and from the corresponding statement for ${\mathtt 1}{\mathtt 0}$ . We see that it is sufficient to write ${\mathbf t}$ as a concatenation of the words

(2-25) $$ \begin{align} w_{{\mathtt a}}:={\mathtt 0}{\mathtt 1}{\mathtt 1},\quad w_{{\bar{\mathtt a}}}:={\mathtt 0}{\mathtt 1}{\mathtt 0},\quad w_{{\mathtt b}}:={\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0},\quad\mbox{and}\quad w_{{\mathtt c}}:={\mathtt 0}{\mathtt 1}, \end{align} $$

since each word $w_x$ takes care of one ${\mathtt 0}{\mathtt 1}$ -block, followed by one ${\mathtt 1}{\mathtt 0}$ -block, and the gap sequence for w is obtained by replacing each $w_x$ by a succession of two gaps. Applying the morphism $\tau $ , we obtain $\tau (w_{{\mathtt a}})=w_{{\mathtt a}}w_{{\bar {\mathtt a}}}$ , $\tau (w_{{\bar {\mathtt a}}})=w_{{\mathtt b}}w_{{\mathtt c}}$ , $\tau (w_{{\mathtt b}})=w_{{\mathtt a}}w_{{\bar {\mathtt a}}}w_{{\mathtt c}}$ , $\tau (w_{{\mathtt c}})=w_{{\mathtt b}}$ . This mimics the morphism $\psi $ ; proceeding as in the proof of Lemma 2.2 (alternatively, as in the proof of Lemma 2.3), we obtain

(2-26) $$ \begin{align} {\mathbf t}=w_{{\bar{\mathbf B}}_0}w_{{\bar{\mathbf B}}_1}w_{{\bar{\mathbf B}}_2}\cdots. \end{align} $$

Since ${\bar {\mathbf B}}$ is morphic, the succession of gaps with which w occurs in ${\mathbf t}$ is morphic by [Reference Allouche and Shallit4, Corollary 7.7.5] (that is, ‘morphic images of morphic sequences are morphic’).

The case $\{a^{{\mathtt 0}{\mathtt 0}}_k,a^{{\mathtt 1}{\mathtt 1}}_k\}$ is similar. Defining

$$ \begin{align*} \tilde w_{{\mathtt a}}:={\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}{\mathtt 1}{\mathtt 0},\quad \tilde w_{{\bar{\mathtt a}}}:={\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}{\mathtt 0}{\mathtt 1},\quad \tilde w_{{\mathtt b}}:={\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}{\mathtt 1}{\mathtt 0}{\mathtt 0}{\mathtt 1},\quad\mbox{and}\quad \tilde w_{{\mathtt c}}:={\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}, \end{align*} $$

it is straightforward to verify that $\tau (\tilde w_{{\mathtt a}})=\tilde w_{{\mathtt a}}\tilde w_{{\bar {\mathtt a}}}$ , $\tau (\tilde w_{{\bar {\mathtt a}}})=\tilde w_{{\mathtt b}}\tilde w_{{\mathtt c}}$ , $\tau (\tilde w_{{\mathtt b}})=\tilde w_{{\mathtt a}}\tilde w_{{\bar {\mathtt a}}}\tilde w_{{\mathtt c}}$ , and $\tau (\tilde w_{{\mathtt c}})=\tilde w_{{\mathtt b}}$ . Again, we can spot the morphism $\psi $ , and we obtain

$$ \begin{align*} {\mathbf t}=\tilde w_{{\bar{\mathbf B}}_0}\tilde w_{{\bar{\mathbf B}}_1}\tilde w_{{\bar{\mathbf B}}_2}\cdots \end{align*} $$

in exactly the same way as before. Each of the words $w_x$ in this representation yields a block ${\mathtt 1}{\mathtt 1}$ in ${\mathbf t}$ , followed by a block ${\mathtt 0}{\mathtt 0}$ . Therefore, also in this case, the gap sequence for w is a morphic image of a morphic sequence. This finishes the proof of Theorem 1.1.

Remark 2.12. Let us have a closer look at the gaps in the ‘mixed case’ $\{a^{{\mathtt 0}{\mathtt 1}}_k,a^{{\mathtt 1}{\mathtt 0}}_k\}$ . Let $\sigma _0$ be the index at which w appears in $a^{{\mathtt 0}{\mathtt 1}}_k$ , and $\sigma _1$ the index at which w appears in $a^{{\mathtt 1}{\mathtt 0}}_k$ . By (2-26) and the choice (2-25), each letter $x\in \{{\mathtt a},{\bar {\mathtt a}},{\mathtt b},{\mathtt c}\}$ in ${\bar {\mathbf B}}$ corresponds to two gaps, as follows.

$$ \begin{align*} \begin{array}{l@{\hspace{2em}}l@{\hspace{2em}}l} \mbox{Letter in } {\bar{\mathbf B}} &\mbox{Gap } 1&\mbox{Gap } 2\\ {\mathtt a}&\sigma_1-\sigma_0+2^{k+1}&\sigma_0-\sigma_1+2^k\\ {\bar{\mathtt a}}& \sigma_1-\sigma_0+2^k&\sigma_0-\sigma_1+2^{k+1}\\ {\mathtt b}& \sigma_1-\sigma_0+2^{k+1}&\sigma_0-\sigma_1+2^{k+1}\\ {\mathtt c}& \sigma_1-\sigma_0+2^k&\sigma_0-\sigma_1+2^k\\ \end{array} \end{align*} $$

It follows that there are at most four gaps that can occur in this case. For example, consider the gap sequence for the factor $w={\mathtt 0}{\mathtt 1}{\mathtt 0}$ . In this case $k=2$ , and we have $a^{{\mathtt 0}{\mathtt 1}}_2={\mathtt 0}{\mathtt 1}{\mathtt 1}\underline {{\mathtt 0}{\mathtt 1}{\mathtt 0}}{\mathtt 0}{\mathtt 1}$ and $a^{{\mathtt 1}{\mathtt 0}}_2={\mathtt 1}{\mathtt 0}\underline {{\mathtt 0}{\mathtt 1}{\mathtt 0}}{\mathtt 1}{\mathtt 1}{\mathtt 0}$ , where the occurrences of w are underlined. We have $\sigma _0=3$ and $\sigma _1=2$ . This yields the gaps $3$ , $5$ , $7$ , and $9$ , occurring only in the combinations $(7,5)$ , $(3,9)$ , $(7,9)$ , and $(3,5)$ . Noting also the first occurrence ${\mathbf t}_3{\mathbf t}_4{\mathbf t}_5={\mathtt 0}{\mathtt 1}{\mathtt 0}$ , the first few occurrences of ${\mathtt 0}{\mathtt 1}{\mathtt 0}$ in ${\mathbf t}$ are at positions $3$ , $10$ , $15$ , $18$ , and $27$ ; compare (1-2). In particular, the gap sequence is not of the form $2^{\ell }{\mathbf B}$ or $2^{\ell }\check {\mathbf B}$ for some $\ell \geq 0$ , each of which has only three different values.

Similar considerations hold for the case $\{a^{{\mathtt 0}{\mathtt 0}}_k,a^{{\mathtt 1}{\mathtt 1}}_k\}$ . More precisely, let $\sigma _0$ be the index at which w appears in $a^{{\mathtt 1}{\mathtt 1}}_k$ and $\sigma _1$ the index at which w appears in $a^{{\mathtt 0}{\mathtt 0}}_k$ . Each letter occurring in ${\bar {\mathbf B}}$ corresponds to two gaps for w, as follows.

$$ \begin{align*} \begin{array}{l@{\hspace{2em}}l@{\hspace{2em}}l} \mbox{Letter in } {\bar{\mathbf B}}&\mbox{Gap } 1&\mbox{Gap } 2\\ {\mathtt a}&\sigma_1-\sigma_0+4\cdot 2^k&\sigma_0-\sigma_1+2\cdot 2^k\\ {\bar{\mathtt a}}& \sigma_1-\sigma_0+2\cdot 2^k&\sigma_0-\sigma_1+4\cdot 2^k\\ {\mathtt b}& \sigma_1-\sigma_0+4\cdot 2^k&\sigma_0-\sigma_1+4\cdot 2^k\\ {\mathtt c}& \sigma_1-\sigma_0+2\cdot 2^k&\sigma_0-\sigma_1+2\cdot 2^k. \end{array} \end{align*} $$

An example for this case is given by the word ${\mathtt 0}{\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}$ , which is a factor of $a^{{\mathtt 0}{\mathtt 0}}_2={\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}{\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}$ and of $a^{{\mathtt 1}{\mathtt 1}}_2={\mathtt 1}{\mathtt 0}{\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}{\mathtt 0}{\mathtt 1}$ . We have $\sigma _0=1$ and $\sigma _1=3$ , and therefore the gaps $6$ , $10$ , $14$ , and $18$ , which appear as pairs $(18,6)$ , $(10,14)$ , $(18,14)$ , and $(10,6)$ .

3.1 $\mathbf A$ is automatic

It has been known since Berstel [Reference Berstel7] that ${\mathbf A}$ is $2$ -automatic. In this section we re-prove this statement using slightly different notation. Note that we give similar proofs (of Lemmas 2.2 and 2.3) in the first part of this paper. First of all, we recapture Berstel’s $2$ -uniform morphism. Introducing an auxiliary letter ${\bar {\mathtt b}}$ , we have the morphism $\overline \varphi $ as well as the coding $\pi $ :

$$ \begin{align*} \begin{array}{lllll} \overline\varphi:& {\mathtt a}\mapsto {\mathtt a}{\mathtt b},& {\mathtt b}\mapsto{\mathtt c}{\mathtt a},& {\bar{\mathtt b}}\mapsto{\mathtt a}{\mathtt c},& {\mathtt c}\mapsto{\mathtt c}{\bar{\mathtt b}}, \\ \pi:&{\mathtt a}\mapsto{\mathtt a},&{\mathtt b}\mapsto{\mathtt b},&{\bar{\mathtt b}}\mapsto{\mathtt b},&{\mathtt c}\mapsto{\mathtt c}. \end{array} \end{align*} $$

We wish to prove that

(3-1) $$ \begin{align} \pi({\overline{\mathbf A}})={\mathbf A}, \end{align} $$

where ${\overline {\mathbf A}}$ is the fixed point of $\overline \varphi $ starting with ${\mathtt a}$ . For this purpose, we show, by induction on $k\geq 0$ , that the initial segment

$$ \begin{align*}s_k:=\overline\varphi^k({\mathtt a}{\mathtt b}{\mathtt c})\end{align*} $$

of ${\overline {\mathbf A}}$ , of length $3\cdot 2^k$ , is a concatenation of the three words $w_0={\mathtt a}{\mathtt b}{\mathtt c}$ , $w_1={\mathtt a}{\mathtt c}$ , and $w_2={\bar {\mathtt b}}$ . We also call the words $w_j$ ‘base words’ in this context, and the latter statement the ‘concatenation property’. Having proved this property, we use (recall the morphism $\varphi $ defined in (2-1))

$$ \begin{align*} \begin{array}{lll} \pi(\overline\varphi(w_1)) &={\mathtt a}{\mathtt b}{\mathtt c}{\mathtt a}{\mathtt c}{\mathtt b}&=\varphi(\pi(w_1)),\\ \pi(\overline\varphi(w_2)) &={\mathtt a}{\mathtt b}{\mathtt c}{\mathtt b}&=\varphi(\pi(w_1)),\\ \pi(\overline\varphi(w_3)) &={\mathtt a}{\mathtt c}&=\varphi(\pi(w_3)), \end{array} \end{align*} $$

in order to obtain

(3-2) $$ \begin{align} \varphi(\pi(s_k)) =\pi(\overline\varphi(s_k)) \end{align} $$

for all $k\geq 0$ , by concatenation. In other words, $\varphi $ and $\overline {\varphi }$ act in the same way on an initial segment of ${\overline {\mathbf A}}$ of length $3\cdot 2^k$ .

We may also display the relation (3-2) graphically. Define $S=\{s_k:k\geq 0\}\subseteq \{{\mathtt a},{\mathtt b},{\bar {\mathtt b}},{\mathtt c}\}^{\mathbb N}$ and $\Omega =\{{\mathtt a},{\mathtt b},{\mathtt c}\}^{\mathbb N}$ . Then the following diagram is commutative:

Gluing together copies of this diagram, we obtain, for all $\ell \geq 1$ ,

$$ \begin{align*} \varphi^{\ell}(s_0)&=\varphi^{\ell}(\pi(s_0)) = \varphi^{\ell-1}(\pi(\overline\varphi(s_0))) \\&= \varphi^{\ell-2}(\varphi(\pi(\overline\varphi(s_0)))) = \varphi^{\ell-2}(\pi(\overline\varphi^2(s_0))) =\cdots=\pi(\overline\varphi^{\ell}(s_0)). \end{align*} $$

For each index $j\geq 0$ , choose $\ell $ so large that $3\cdot 2^{\ell }\geq j$ . Then

$$ \begin{align*}{\mathbf A}_i=\varphi^{\ell}(s_0)\rvert_i=\pi(\overline\varphi^{\ell}(s_0))\rvert_i= \pi(\overline\varphi^{\ell}(s_0)\rvert_i)= \pi({\overline{\mathbf A}}_i)\end{align*} $$

for $0\leq i<j$ . Therefore the infinite word ${\mathbf A}$ is $2$ -automatic, being the coding under $\pi $ of the $2$ -automatic sequence ${\overline {\mathbf A}}$ , and thus we have derived (3-1) from the concatenation property.

We still have to prove that $s_k$ is a concatenation of the base words. Clearly, this holds for $s_0={\mathtt a}{\mathtt b}{\mathtt c}=w_0$ . Assume that we have already established that $s_k=w_{\varepsilon _0}w_{\varepsilon _1}\cdots $ for some $\varepsilon _j\in \{0,1,2\}$ . We have

$$ \begin{align*} \overline\varphi(w_1)&={\mathtt a}{\mathtt b}{\mathtt c}{\mathtt a}{\mathtt c}{\bar{\mathtt b}}=w_0w_1w_2,\\ \overline\varphi(w_2)&={\mathtt a}{\mathtt b}{\mathtt c}{\bar{\mathtt b}}=w_0w_2,\\ \overline\varphi(w_3)&={\mathtt a}{\mathtt c}=w_1, \end{align*} $$

and thus

$$ \begin{align*}s_{k+1}=\overline\varphi(s_k) =\overline\varphi(w_{\varepsilon_0})\overline\varphi(w_{\varepsilon_1})\cdots\end{align*} $$

is a concatenation of the $w_j$ too. This proves (3-1).

Complementing this result, we note that Berstel [Reference Berstel7, Corollary 7] also proved that ${\mathbf A}$ itself is not a fixed point of (the extension of) a uniform morphism.

3.2 Transforming $\mathbf A$

We identify circular shifts, or rotations, of factors of length $L\geq 2$ appearing in the sequence ${\mathbf A}$ . Such a rotation of a word $(a_i)_{i\geq 0}$ replaces the subword $a_j\,a_{j+1}\cdots a_{j+L-2}\,a_{j+L-1}$ by $a_{j+1}\cdots a_{j+L-2}\,a_{j+L-1}\,a_j$ (rotation to the left), or $a_{j+L-1}\,a_j\,a_{j+1}\cdots a_{j+L-2}$ (rotation to the right).

Carrying out a certain number of such rotations, we see that the sequence $\mathbf A$ is reduced to the periodic word $({\mathtt a}{\mathtt b}{\mathtt c})^{\omega }$ . Of course, this is possible for any word containing an infinite number of each of ${\mathtt a}$ , ${\mathtt b}$ , and ${\mathtt c}$ , and it can be achieved in uncountably many ways. In our case, however, an admissible sequence of rotations can be made very explicit, by defining a new morphism $\varphi ^+$ . This morphism has the fixed point ${\overline {\mathbf A}}$ , which maps to ${\mathbf A}$ under a coding. From this augmented sequence, we see very clearly the ‘nested structure’ of the above-mentioned rotations. In particular, we can find a certain noncrossing matching, defined in (3-8), describing the intervals that we perform rotations on, and the direction of each rotation. Moreover, in the process we learn something about the discrepancy of ${\mathtt 0}{\mathtt 1}$ -blocks in ${\mathbf t}$ , which was defined in (1-4). Let us consider the iteration $\overline \varphi ^2$ of Berstel’s morphism:

$$ \begin{align*} \begin{array}{lllll} \overline\varphi^2:& {\mathtt a}\mapsto {\mathtt a}{\mathtt b}{\mathtt c}{\mathtt a},& {\mathtt b}\mapsto{\mathtt c}{\bar{\mathtt b}}{\mathtt a}{\mathtt b},& {\bar{\mathtt b}}\mapsto{\mathtt a}{\mathtt b}{\mathtt c}{\bar{\mathtt b}},& {\mathtt c}\mapsto{\mathtt c}{\bar{\mathtt b}}{\mathtt a}{\mathtt c}. \end{array} \end{align*} $$

We introduce certain decorations (connectors) of the letters. Their meaning becomes clear in a moment. Based on the morphism $\overline \varphi ^2$ , we define the following decorated version, which is a morphism on the seven-letter alphabet

(3-3)

This morphism has a unique fixed point ${{\mathbf A}^{\!+}}$ starting with ${\mathtt a}$ . The image of ${{\mathbf A}^{\!+}}$ under the obvious coding $\gamma $ given by

(3-4)

yields the sequence ${\mathbf A}$ . Based on this, we speak of letters of types ${\mathtt a}$ , ${\mathtt b}$ , and ${\mathtt c}$ , thus referring to letters from $\{{\mathtt a}\}$ , , and , respectively.

From substitution (3-3), we can immediately derive the following lemma.

Lemma 3.1. Let $j\geq 1$ , and $(x,y,z)=({{\mathbf A}^{\!+}}_{j-1},{{\mathbf A}^{\!+}}_j,{{\mathbf A}^{\!+}}_{j+1})$ . Then

(3-5)

We wish to connect the ‘loose ends’ of the connectors; we say that two connectors at indices $i<j$ match if the connector at i points to the right and the connector at j points to the left. The very simple algorithm FindMatching joins matching connectors, beginning with shortest connections. Only pairs of free connectors are connected, that is, each letter may be the starting point of only one link.

Note that we have to pay attention that previously selected indices are not chosen again. This explains the introduction of SelectedIndices. A connection between the two letters at indices i and j is just a different name for the pair $(i,j)$ . For any finite word w over the alphabet K this procedure yields a (possibly empty) set $M(w)$ of pairs $(i,j)$ of indices.

We wish to prove that the algorithm is monotone.

Lemma 3.2. Let w and $w'$ be finite words over the alphabet K, and assume that w is an initial segment of $w'$ . Let $M(w)$ (respectively, $M(w')$ ) be the sets of pairs found by the FindMatching algorithm. Then

$$ \begin{align*} M(w)\subseteq M(w'). \end{align*} $$

We extend M to a function on all (finite or infinite) words w over K, in the following obvious way: for each $\ell $ , form the set $\tilde M_{\ell }(w)$ of all pairs $(a,b)$ satisfying $b-a=\ell $ , having matching connectors, such that neither a nor b is a component of any $\tilde M_{\ell '}(w)$ , where $\ell '<\ell $ . Set $\tilde M(w)=\bigcup _{\ell \geq 1}\tilde M_{\ell }(w)$ . The following lemma gives us a method to compute a matching for an infinite word by only looking at finite segments.

Lemma 3.3. Let w be an infinite word over K. Then

(3-6) $$ \begin{align} \bigcup_{j\geq 0}M(w\rvert_{[0,j)})=\tilde M(w). \end{align} $$

Proof. Let $M_{\ell }(w)$ be the set of connections added in step $\ell $ of the FindMatching algorithm. We prove, more generally, that

(3-7) $$ \begin{align} \bigcup_{j\geq 0}M_{\ell}(w\rvert_{[0,j)})=\tilde M_{\ell}(w). \end{align} $$

We prove this by induction on $\ell $ , and we start at connections of length $\ell =1$ . Let $(i,i+1)\in M_{\ell }(w\rvert _{[0,j)})$ . Then there is a pair of matching connectors at indices i and $i+1$ (where $i+1<j$ ), and therefore this pair is also contained in $\tilde M_1(w)$ . This proves the inclusion ‘ $\subseteq $ ’. On the other hand, if $(i,i+1)$ is a link connecting matching connectors in w, this link is also to be found in the sequence $w\rvert _{[0,i+2)}$ , hence the inclusion ‘ $\supseteq $ ’. Assume that (3-7) holds for some $\ell \geq 1$ . If the algorithm finds a pair $(i,i+\ell )$ of matching connectors in $w\rvert _{[0,j)}$ , where $(i,i+\ell )\not \in M_{\ell }(w\rvert _{[0,j)})$ , this pair trivially also matches in the (unrestricted) word w. By hypothesis, the connectors at i and $i+\ell $ are not used by $\tilde M_{\ell }(w)$ , hence the inclusion ‘ $\subseteq $ ’. On the other hand, a link $(i,i+\ell )$ of matching connectors in w that is still free in step $\ell $ is also free in $w\rvert _{[0,i+\ell +1)}$ by hypothesis, which proves (3-7) and hence the lemma.

Our algorithm avoids crossing connections: if $i<j<k<\ell $ are indices such that $(i,k)\in M(w)$ and $({\kern1.5pt}j,\ell )\in M(w)$ , then the connector at index j is pointing to the right, and the one at k to the left, so the shorter connection $({\kern1.5pt}j,k)$ would have been chosen earlier. This contradicts the construction rule that indices may only be chosen once.

More generally, a noncrossing matching for a word w over K (finite or infinite) is a set M of pairs $(i,j)$ such that

(3-8)

Here $\bigcup M=\{i:(i,j)\in M\mbox { for some }j\ \mbox {or }({\kern1.5pt}j,i)\in M\mbox { for some }j\}$ .

We call a word w closed if there exists a noncrossing matching for w.

Proof. First of all, we note that it is sufficient to prove that $\varphi ^+$ maps closed words w to closed words. If this is established, we obtain, by induction, that the initial segments $(\varphi ^+)^k({\mathtt a})$ of ${{\mathbf A}^{\!+}}$ are closed. Since noncrossing matchings are unique, the corresponding sequence $(m_k)_{k\geq 0}$ of noncrossing matchings satisfies $m_k\subseteq m_{k+1}$ , and $\bigcup _{k\geq 0}m_k$ is easily seen to be the desired matching for ${{\mathbf A}^{\!+}}$ .

We prove by induction on the length n of a closed word w that $\varphi ^+(w)$ is closed. This is obvious for the closed words of length $n\le 2$ : the word is closed, and the cases , , , and are also easy. Moreover, a concatenation of two closed words is also closed: one of the matchings has to be shifted (both components of each entry have to be shifted), and we only have to form the union of the matchings.

If w is of the form for some nonempty word C over K, we obtain a noncrossing matching for C by stripping the pair $(1,n)$ from a corresponding matching for w. Therefore, C is closed. Applying $\varphi ^+$ , we see that

(3-9)

This is closed by our hypothesis, since C is shorter than w. The other case is analogous (note that there are no more cases by (3-5)), and the proof is complete.

The construction of the matching in the proof of Lemma 3.5 also shows the following result.

Our interest in the link structure of ${{\mathbf A}^{\!+}}$ stems from the fact that we may transform the sequence ${\mathbf A}$ into a periodic one, using the following transparent mechanism. Let m be the noncrossing matching for ${{\mathbf A}^{\!+}}$ , and let $((i_k,j_k))_{k\geq 0}$ be an enumeration of m such that $({\kern1.5pt}j_k-i_k)_{k\geq 0}$ is nondecreasing. We define a sequence $(A^{(k)})_{k\geq 0}$ as follows.

• • Set $A^{(0)}={{\mathbf A}^{\!+}}$ .

• • Let $k\geq 0$ . If , we rotate the letters in $A^{(k)}$ with indices $i_k,i_k+1,\ldots ,j_k$ to the right by one place, yielding $A^{(k+1)}$ . Otherwise, we necessarily have and we rotate the letters with indices $i_k,i_k+1,\ldots ,j_k-1$ to the left by one place.

In more colourful language, in each step some letter of type ${\mathtt b}$ is moved along its connecting link and inserted just before the letter of type ${\mathtt c}$ it is connected to. Note that due to the monotonicity requirement and the noncrossing property, the $k\, $ th rotation does not change the indices at which the subsequent rotations are carried out. Therefore, the sequence $(A^{(k)})_{k\ge 0}$ is well defined. Moreover, the result does not depend on the particular nondecreasing enumeration of m for the same reasons. Since the first N indices eventually remain unchanged, the limit

$$ \begin{align*} \rho({{\mathbf A}^{\!+}}):=\gamma\big(\lim_{k\rightarrow\infty}A^{(k)}\big) \end{align*} $$

exists (note that $\gamma $ , defined in (3-4), replaces each letter of type x by x). The definition of $A^{(k)}$ is summarized in the RotateAlongLinks algorithm. As in the case of the FindMatching algorithm, we require a finite word w (and a finite set $m\subset \mathbb N^2$ ) as input in order to guarantee finite running time.

By the above remarks, the words RotateAlongLinks $(w_k,M(w_k))$ converge to $\rho ({{\mathbf A}^{\!+}})$ as $k\rightarrow \infty $ , where $w=(\varphi ^+)^k({\mathtt a})$ . We have the following central proposition.

Proposition 3.8. Let m be the noncrossing matching for ${{\mathbf A}^{\!+}}$ . Then

$$ \begin{align*} \rho({{\mathbf A}^{\!+}})=({\mathtt a}{\mathtt b}{\mathtt c})^{\omega}. \end{align*} $$

Proof. Let us first note that the limit itself can be obtained in a simpler way. For any closed word C over K, (1) apply $\gamma $ , (2) remove all occurrences of ${\mathtt b}$ , and (3) reinsert ${\mathtt b}$ before each ${\mathtt c}$ . The resulting word equals $\rho (C)$ . This statement simply follows from the facts that (i) both procedures do not change the order in which the underlying letters ${\mathtt a}$ and ${\mathtt c}$ appear, that (ii) each occurrence of ${\mathtt c}$ in both results is preceded by ${\mathtt b}$ , and that (iii) in both results, ${\mathtt b}$ does not appear at other places. We therefore see that Proposition 3.8 is equivalent to the following. Let $\mathbf C$ be the sequence obtained from ${{\mathbf A}^{\!+}}$ by deleting all decorations, and all occurrences of ${\mathtt b}$ and ${\bar {\mathtt b}}$ . Then $\mathbf C=({\mathtt a}{\mathtt c})^{\omega }$ . In other words, we only have to show that ${\mathtt a}$ and ${\mathtt c}$ occur alternatingly in ${\mathbf A}$ , with the empty word or one occurrence of ${\mathtt b}$ in between. We prove a stronger statement concerning the sequence ${\overline {\mathbf A}}$ , which completes the proof of Proposition 3.8.

Lemma 3.9. There are sequences $(\varepsilon _k)_{k\geq 0}$ and $(\varepsilon ^{\prime }_k)_{k\geq 0}$ in $\{0,1\}$ such that

$$ \begin{align*}{\overline{\mathbf A}}={\mathtt a} ({\mathtt b}{\mathtt c}({\mathtt a}{\mathtt c})^{\varepsilon_0}{\bar{\mathtt b}}{\mathtt a}({\mathtt c}{\mathtt a})^{\varepsilon^{\prime}_0}) ({\mathtt b}{\mathtt c}({\mathtt a}{\mathtt c})^{\varepsilon_1}{\bar{\mathtt b}}{\mathtt a}({\mathtt c}{\mathtt a})^{\varepsilon^{\prime}_1}) \cdots. \end{align*} $$

In order to prove this, we apply the second iteration $\overline \varphi ^2$ of Berstel’s morphism to one of the expressions in brackets. We use the abbreviation

$$ \begin{align*}b(\varepsilon,\varepsilon')={\mathtt b}{\mathtt c}({\mathtt a}{\mathtt c})^{\varepsilon}{\bar{\mathtt b}}{\mathtt a}({\mathtt c}{\mathtt a})^{\varepsilon'}.\end{align*} $$

Direct computation yields

$$ \begin{align*} \begin{aligned} \overline\varphi^2(b(0,0)) &={\mathtt c}{\bar{\mathtt b}}{\mathtt a}{\mathtt b}{\mathtt c}{\bar{\mathtt b}}{\mathtt a}{\mathtt c}{\mathtt a}{\mathtt b}{\mathtt c}{\bar{\mathtt b}}{\mathtt a}{\mathtt b}{\mathtt c}{\mathtt a} ={\mathtt c}{\bar{\mathtt b}}{\mathtt a}\,b(0,1)b(0,0){\mathtt b}{\mathtt c}{\mathtt a},\\ \overline\varphi^2(b(0,1)) &={\mathtt c}{\bar{\mathtt b}}{\mathtt a}{\mathtt b}{\mathtt c}{\bar{\mathtt b}}{\mathtt a}{\mathtt c}{\mathtt a}{\mathtt b}{\mathtt c}{\bar{\mathtt b}}{\mathtt a}{\mathtt b}{\mathtt c}{\mathtt a}{\mathtt c}{\bar{\mathtt b}}{\mathtt a}{\mathtt c}{\mathtt a}{\mathtt b}{\mathtt c}{\mathtt a} ={\mathtt c}{\bar{\mathtt b}}{\mathtt a}\,b(0,1)b(0,0)b(1,1){\mathtt b}{\mathtt c}{\mathtt a},\\ \overline\varphi^2(b(1,0)) &={\mathtt c}{\bar{\mathtt b}}{\mathtt a} {\mathtt b}{\mathtt c}{\bar{\mathtt b}}{\mathtt a}{\mathtt c}{\mathtt a}{\mathtt b}{\mathtt c}{\mathtt a}{\mathtt c}{\bar{\mathtt b}}{\mathtt a}{\mathtt c}{\mathtt a}{\mathtt b}{\mathtt c}{\bar{\mathtt b}}{\mathtt a}{\mathtt b}{\mathtt c}{\mathtt a} ={\mathtt c}{\bar{\mathtt b}}{\mathtt a}\,b(0,1)b(1,1)b(0,0){\mathtt b}{\mathtt c}{\mathtt a},\\ \overline\varphi^2(b(1,1)) &={\mathtt c}{\bar{\mathtt b}}{\mathtt a} {\mathtt b}{\mathtt c}{\bar{\mathtt b}}{\mathtt a}{\mathtt c}{\mathtt a}{\mathtt b}{\mathtt c}{\mathtt a}{\mathtt c}{\bar{\mathtt b}}{\mathtt a}{\mathtt c}{\mathtt a}{\mathtt b}{\mathtt c}{\bar{\mathtt b}}{\mathtt a}{\mathtt b}{\mathtt c}{\mathtt a}{\mathtt c}{\bar{\mathtt b}}{\mathtt a}{\mathtt c}{\mathtt a}{\mathtt b}{\mathtt c}{\mathtt a} \end{aligned}\\ ={\mathtt c}{\bar{\mathtt b}}{\mathtt a}\,b(0,1)b(1,1)b(0,0)b(1,1){\mathtt b}{\mathtt c}{\mathtt a}. \end{align*} $$

Arbitrary concatenations of these expressions are again of the form ${\mathtt c}{\bar {\mathtt b}}{\mathtt a} R\,{\mathtt b}{\mathtt c}{\mathtt a}$ , where R is a concatenation of words $b(\varepsilon ,\varepsilon ')$ . Assuming that w is of the form ${\mathtt a}\prod _{j<r}b(\varepsilon _j,\varepsilon ^{\prime }_j){\mathtt b}{\mathtt c}{\mathtt a}$ , we obtain $\overline \varphi ^2(w)={\mathtt a}\,b(1,0)R\,{\mathtt b}{\mathtt c}{\mathtt a}$ . Since the words $(\overline \varphi ^2)^k({\mathtt a})$ approach a fixed point, and

$$ \begin{align*}\overline\varphi^4({\mathtt a})={\mathtt a}\,b(1,0)b(0,1){\mathtt b}{\mathtt c}{\mathtt a},\end{align*} $$

it follows by induction that ${\overline {\mathbf A}}$ is indeed of the form stated in the lemma, and we have, in particular, proved Proposition 3.8.

From this algorithm, we can clearly see that a given letter ${\mathtt a}$ is shifted, one place at a time, for each link that is passing over this letter. The direction in which ${\mathtt a}$ is shifted depends on whether or appears in the link we are dealing with. We use considerations of this kind in the following section, together with Proposition 3.8, in order to determine the discrepancy of ${\mathtt 0}{\mathtt 1}$ -occurrences in ${\mathbf t}$ .

3.3 The discrepancy of ${\mathtt 0}{\mathtt 1}$ -blocks

For an integer $j\geq 0$ let us define the degree of j as follows. Let m be the noncrossing matching for ${{\mathbf A}^{\!+}}$ and set

We also talk about the degree of a letter in ${{\mathbf A}^{\!+}}$ , where the position in question will always be clear from the context.

We display the first $192$ letters of ${{\mathbf A}^{\!+}}$ , obtained by applying the third iteration of $\varphi ^+$ to the word

, and we connect associated connectors by actual lines for better readability. In position $10=({\mathtt 2}{\mathtt 2})_4$ in ${{\mathbf A}^{\!+}}$ , we have a letter ${\mathtt a}$ of degree $-1$ , and in position $170=({\mathtt 2}{\mathtt 2}{\mathtt 2}{\mathtt 2})_4$ , a letter ${\mathtt a}$ of degree $-2$ . These positions are marked with an arrow.

(3-10)

From this initial segment we see that the sequence $(\deg ({\kern1.5pt}j))_{j\geq 0}$ starts with the $48$ integers

$$ \begin{align*}\begin{aligned} &0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0, \\&-1,0,0,0,0,0,-1,-1,-1,-1,0,0,-1,0,\ldots, \end{aligned} \end{align*} $$

corresponding to the first line of (3-10). Applying the substitution $(\varphi ^+)^2$ to

appearing in positions $169$ – $171$ , we obtain the following $48$ letters. The marked letter ${\mathtt a}$ has degree $-3$ , and it corresponds to the position $({\mathtt 2}{\mathtt 2}{\mathtt 2}{\mathtt 2}{\mathtt 2}{\mathtt 2})_4$ .

In general, in position $({\mathtt 2}^{2k})_4$ , a letter ${\mathtt a}$ of degree $-k$ appears. This can be seen by considering the images of ${\mathtt a}$ and

under $\varphi ^+$ .

By Proposition 3.8, $\deg ({\kern1.5pt}j)$ has the following meaning in the case where ${{\mathbf A}^{\!+}}_j={\mathtt a}$ . A number of letters (of which there are $\mathrm{deg}^+({\kern1.5pt}j)$ ) are transferred from the right of the letter ${\mathtt a}$ to the left of it; note that the letter ${\mathtt a}$ is shifted to the right $\deg ^+({\kern1.5pt}j)$ places. Analogously, $\deg ^-({\kern1.5pt}j)$ letters are transferred from the left of ${\mathtt a}$ to the right, and the letter ${\mathtt a}$ is shifted to the left $\deg ^-({\kern1.5pt}j)$ places. In total, the letter ${\mathtt a}$ (among other letters) is shifted by $\deg ({\kern1.5pt}j)$ places, and ${\mathtt b}$ s or ${\bar {\mathtt b}}$ s are moved to account for the generated trailing space. The proposition states that the letters to the left of ${\mathtt a}$ ’s new position $j+\deg ^+({\kern1.5pt}j)$ are balanced: after removing decorations and replacing ${\bar {\mathtt b}}$ by ${\mathtt b}$ , the letters ${\mathtt a}$ , ${\mathtt b}$ , and ${\mathtt c}$ occur the same number of times. If , similar considerations hold. The case of letters of type ${\mathtt b}$ is different, since a single rotation may shift such a letter to a remote place.

The transducer $\mathcal T_1$ displayed in Figure 1 allows us to compute the degree of an arbitrary position j: starting from the centre node, we traverse the graph, guided by the base- $4$ expansion $\delta _{\nu -1}\cdots \delta _0$ of j (read from left to right). Along the way, we sum up the numbers k whenever a vertex $\delta _i\mid k$ is taken. The sum over these numbers is the degree of j, multiplied by $3$ . The transducer $\mathcal T_1$ is derived directly from the decorated, $4$ -uniform morphism $\varphi ^+$ given in (3-3). Note that a change of degree takes place whenever new letters are inserted, by virtue of the morphism $\varphi ^+$ , into the range of already existing links, which happens for and ; or if a new link together with a letter ${\mathtt a}$ in its range is created, which happens for .

Figure 1

A base- $4$ transducer that generates the degree sequence.

We now apply Proposition 3.8 to the discrepancy $D_N$ of occurrences of ${\mathtt 0}{\mathtt 1}$ in ${\mathbf t}$ .

Proposition 3.10. Let $j\in \mathbb N$ and set $d=\deg ({\kern1.5pt}j)$ . Then

(3-11)

In each of these cases, the subscript of D is the position in ${\mathbf t}$ that corresponds to the $j\, $ th letter in ${\mathbf A}$ via (2-3).

Proof. Choose $\varepsilon \in \{0,1,2\}$ and $n\in \mathbb N$ such that $j=3n+\varepsilon $ . Let us consider each of the seven cases corresponding to letters from K.

First case. Assume that ${{\mathbf A}^{\!+}}_j={\mathtt a}$ . By the RotateAlongLinks algorithm and Proposition 3.8, a total of d letters of type ${\mathtt b}$ have to be shifted from the right of our ${\mathtt a}$ in question to the left (if $d>0$ ), or the other way round (if $d<0$ ). After this procedure, the numbers of letters of types ${\mathtt a}$ , ${\mathtt b}$ , and ${\mathtt c}$ to the left are equal. It follows that $\varepsilon \equiv -d\bmod 3$ ; moreover, $m=n+(\varepsilon +d)/3$ is the number of letters ${\mathtt a}$ (and also the number of letters of type ${\mathtt c}$ ) strictly to the left of j. The number of letters of type ${\mathtt b}$ to the left of j is $m'=n+(\varepsilon -2d)/3$ . Symbols of type ${\mathtt a}$ contribute two blocks ${\mathtt 0}{\mathtt 1}$ and correspond to a factor of length $6$ in ${\mathbf t}$ , by (2-2); letters of type ${\mathtt b}$ contribute one block and correspond to a factor of length $4$ ; letters of type ${\mathtt c}$ contribute one block and correspond to a factor of length $2$ . It follows that below position

$$ \begin{align*} N=(6+2)\bigg(n+\frac{\varepsilon+d}3\bigg)+4 \bigg(n+\frac{\varepsilon-2d}{3}\bigg) =12n+4\,\varepsilon=4j \end{align*} $$

we find

$$ \begin{align*}(2+1)\bigg(n+\frac{\varepsilon+d}3\bigg)+\bigg(n+\frac{\varepsilon-2d}3\bigg) =4n+4\,\varepsilon/3+d/3 \end{align*} $$

${\mathtt 0}{\mathtt 1}$ -blocks. This proves the case ${{\mathbf A}^{\!+}}_j=a$ .

Second case. If , we note that necessarily ${{\mathbf A}^{\!+}}_{j+1}={\mathtt a}$ , by Lemma 3.1. We apply the first case in position $j+1$ , which has degree d. Noting that a letter of type ${\mathtt b}$ in ${{\mathbf A}^{\!+}}$ corresponds to ${\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}$ in Thue–Morse, we obtain $D_{4j}=D_{4j+4}+1/3=d/3+1/3$ , where $4j$ (respectively, $4j+2$ ) corresponds to the $j\, $ th (respectively, $({\kern1.5pt}j+1)\, $ th) position in ${{\mathbf A}^{\!+}}$ .

Third case. Assume that . In this case, the letter at $j+1$ is ${\mathtt a}$ by Lemma 3.1 and $j+1$ has degree $d-1$ . It follows that $D_{4j}=D_{4j+4}+1/3=d/3-1/3+1/3=d/3$ .

Fourth case. If , we note that necessarily ${{\mathbf A}^{\!+}}_{j-1}={\mathtt a}$ ; we apply the first case in position $j-1$ , which has degree $d+1$ . Since ${\mathtt a}$ corresponds to ${\mathtt 0}{\mathtt 1}{\mathtt 1}{\mathtt 0}{\mathtt 1}{\mathtt 0}$ in Thue–Morse, we have $D_{4j+2}=D_{4j-4}=d/3+1/3$ , where $4j-4$ (respectively, $4j+2$ ) corresponds to the $({\kern1.5pt}j-1)\, $ th (respectively, $j\, $ th) positions in ${{\mathbf A}^{\!+}}$ .

Fifth case. Assume that . Then ${{\mathbf A}^{\!+}}_{j-1}={\mathtt a}$ , and $j-1$ has degree d. Analogously to the fourth case, we obtain $D_{4j+2}=D_{4j-4}=d/3$ .

Sixth case. If , this letter is connected to a letter of type ${\mathtt b}$ to the left, which stays on the left of after applying the rotations. Therefore, the number of letters of type ${\mathtt b}$ to the left is changed by d, and the numbers of letters of types ${\mathtt a}$ or ${\mathtt c}$ to the left stay the same. Similarly to the first case, it follows that $\varepsilon \equiv 2-d\bmod 3$ . The numbers of letters, of type ${\mathtt a}$ , ${\mathtt b}$ , and ${\mathtt c}$ , to the left of j, are therefore $m=n+(\varepsilon +d+1)/3$ , $m-d$ , and $m-1$ , respectively. It follows that, below position

$$ \begin{align*} N&=6\bigg(n+\frac{\varepsilon+d+1}3\bigg) +4\bigg(n+\frac{\varepsilon-2d+1}3\bigg) +2\bigg(n+\frac{\varepsilon+d-2}3\bigg) \\&=12n+4\,\varepsilon+2=4j+2 , \end{align*} $$

there are

$$ \begin{align*}2\bigg(n+\frac{\varepsilon+d+1}3\bigg) +\bigg(n+\frac{\varepsilon-2d+1}3\bigg) +\bigg(n+\frac{\varepsilon+d-2}3\bigg) \!=4n+\frac{4\,\varepsilon}3+\frac d3+\frac13 \end{align*} $$

${\mathtt 0}{\mathtt 1}$ -blocks.

Seventh case. If , a letter of type ${\mathtt b}$ takes its place after one rotation. In this case, we have $\varepsilon \equiv 1-d\bmod 3$ ; the numbers of letters to the left of j, of types ${\mathtt a}$ , ${\mathtt b}$ , and ${\mathtt c}$ , are therefore $m=n+(\varepsilon +d+2)/3$ , $m-d-1$ , and $m-1$ , respectively. Therefore, below position

$$ \begin{align*} N&=6\bigg(n+\frac{\varepsilon+d+2}3\bigg) +4\bigg(n+\frac{\varepsilon-2d-1}3\bigg) +2\bigg(n+\frac{\varepsilon+d-1}3\bigg) \\&=12n+4\,\varepsilon+2=4j+2 , \end{align*} $$

there are

$$ \begin{align*}2\bigg(n+\frac{\varepsilon+d+2}3\bigg) +\bigg(n+\frac{\varepsilon-2d-1}3\bigg) +\bigg(n+\frac{\varepsilon+d-1}3\bigg) \!=4n+\frac{4\,\varepsilon}3+\frac d3+\frac23 \end{align*} $$

${\mathtt 0}{\mathtt 1}$ -blocks, which proves the last case.