Proof of part (i) of Theorem 1.2, assuming part (ii)

We will have to confirm that the conditions of Lemma 3.2 are satisfied. Let $a_1,\ldots ,a_k\in \mathcal {H}$ be as in the statement of Theorem 1.2. Condition (a) of Lemma 3.2 is satisfied by our hypothesis. Now, we verify the second condition.

Fix some $i\in \{1,2,\ldots ,k\}$ . We consider three cases.

1. (i) Assume that the function $a_i(t)$ is such that $|a_i(t)-q(t)|\succ t^{\varepsilon }$ for all polynomials $q(t)$ with rational coefficients. Then, the sequence $a_i(n)\mathbb {Z}$ is equidistributed on $\mathbb {T}$ (satisfying condition (1)), due to Theorem A.

2. (ii) Assume that the function $a_i(t)$ is such that $\lim \nolimits _{t\to +\infty } a_i(t)=c\notin \mathbb {Z}$ . Then, the sequence $a_i(n)$ satisfies condition (2) of Lemma 3.2.

3. (iii) Assume that neither of the above conditions is true. Since $a_i(t)$ must satisfy equation (9), we deduce that $a_i(t)$ converges to some integer c. However, since $a_i(t)$ converges to c monotonically (functions in $\mathcal {H}$ are eventually monotone), we deduce that condition (3) of Lemma 3.2 is satisfied and we are done.

Proof that Proposition 4.1 implies Theorem 1.2

We want to show that the averages

$$ \begin{align*} \frac{1}{N}\sum_{i=1}^{N} f_1(b_1^{a_1(n)}x_1) \cdots f_k(b_k^{a_k(n)}x_k) \end{align*} $$

converge for all $x_i\in X_i$ , where the nilmanifolds $X_i=G_i/\Gamma _i$ , the elements $b_i$ and the functions $a_i\in \mathcal {H}$ are as in the statement of part (ii) of Theorem 1.2. We define the continuous function F on the product nilmanifold $X_1\times \cdots \times X_k$ by the relation

$$ \begin{align*}F(y_1,\ldots,y_k)=f_1(y_1)\cdots f_k(y_k).\end{align*} $$

We also denote by $\widetilde {b_i}$ the element on $G_1\times \cdots \times G_k$ , whose ith coordinate is equal to $b_i$ , while all of its other coordinates are equal to the respective identity element. Observe that the elements $\widetilde {b_1},\ldots ,\widetilde {b_k}$ are pairwise commuting. Finally, let us also denote by x the point $(x_1,\ldots ,x_k)$ on the product $X_1\times \cdots \times X_k$ . Then, a simple computation implies that our initial average is equal to

$$ \begin{align*} \underset{1\leq n\leq N}{\operatorname*{\mathrm{\mathbb{E}}}} F(b_1^{a_1(n)}\cdots b_k^{a_k(n)} x) \end{align*} $$

and the claim now follows.

Proof that Lemma 4.2 implies Proposition 4.1

Applying Lemma A.5, we can find a basis ${f_1,\ldots ,f_s}$ for the set $\mathcal {L}(a_1,\ldots ,a_k)$ of non-trivial linear combinations. The collection of functions $f_1,\ldots ,f_s$ can be written in the form $(g_1,\ldots ,g_m,h_1,\ldots ,h_{\ell })$ where $g_i,h_{i}$ are as in Lemma A.5. We will not use this specific property until a little further below, so as to avoid cumbersome notation. Note that the fact that $f_1,\ldots ,f_s$ form a basis indicates that the assumptions on the linear combinations of the $a_1,\ldots ,a_k$ in the statement of Proposition 4.1 are now transferred to the functions $f_1,\ldots ,f_s$ .

If we write

(21) $$ \begin{align} a_i(t)=\sum_{j=1}^{s} c_{i,j}f_j(t) \end{align} $$

for some real numbers $c_{i,j}$ , then we can rewrite the average in equation (18) as

(22) $$ \begin{align} \underset{1\leq n\leq N}{\operatorname*{\mathrm{\mathbb{E}}}} F(u_1^{f_1(n)}\cdots u_s^{f_s(n)}x) \end{align} $$

for some commuting elements $u_1,\ldots ,u_s\in G$ (here, the fact that the elements $b_1,\ldots ,b_k$ commute is required). We denote

$$ \begin{align*}v(n)=u_1^{f_1(n)}\cdots u_s^{f_s(n)},\end{align*} $$

which is a sequence in G. We want to establish that the averages of the sequence $F(v(n)x)$ converge for all $x\in X$ and any continuous function F. If one of the functions $f_1,\ldots ,f_m$ is such that the limit $\lim \nolimits _{t\to +\infty } f_i(t)$ is a real number (which can be the case when a linear combination of the original functions satisfies equation (9)), we can invoke the continuity of F to eliminate the corresponding term $u_i^{f_i(n)}$ in the product and replace it by a constant. Hence, we may assume that all of the functions $f_1(t),\ldots ,f_s(t)$ go to $\pm \infty $ , as $t\to +\infty $ .

Now we use the particular structure of the functions $f_1,\ldots ,f_s$ . The statement of Lemma A.5 implies that the collection of functions $f_1,\ldots ,f_s$ has the form $(g_1,\ldots ,g_m,h_1,\ldots ,h_{\ell })$ (clearly, $m+\ell =s$ ) such that the functions $g_i$ can be written in the form $p_i(t)+x_i(t)$ , where the functions $x_1(t),\ldots ,x_{m}(t)$ are strongly non-polynomial and have pairwise distinct (and non-trivial) growth rates, while the functions $h_i$ can be written in the form $\widetilde {p}_i(t)+y_i(t)$ , where $y_i(t)$ converges to 0. Here, $p_i$ and $\widetilde {p}_i$ are polynomials with real coefficients.

We may rearrange the functions $f_i$ so that $f_i=g_i$ for all $1\leq i\leq m$ and $f_j=h_{j-m}$ for each $m+1\leq j\leq s$ . Rewrite the sequence $v(n)$ as

$$ \begin{align*} v(n)=\prod_{i=1}^{m}u_i^{g_i(n)}\cdot \prod_{i=1}^{\ell} u_{m+i}^{h_i(n)}=\prod_{i=1}^{m}u_i^{p_i(n)+x_i(n)}\cdot \prod_{i=1}^{\ell} w_i^{\widetilde{p}_i(n)+y_i(n)}, \end{align*} $$

where we use the notation $w_i$ for the element $u_{i+m}$ in the last equality. Without loss of generality, assume that

$$ \begin{align*}x_1(t)\succ x_2(t)\succ \cdots \succ x_{m}(t)\succ 1.\end{align*} $$

First, we need to distinguish between the sub-fractional functions and the ‘fast’ growing functions among the functions $x_i(t)$ (this will be important later when we use the polynomial expansion). Thus, let $0\leq s_0\leq m$ be a natural number such that $x_{s_0}(t)\gg t^{\varepsilon }$ for some $\varepsilon>0$ , while $x_{s_0+1}$ is a sub-fractional function. This also implies that all the functions $x_{i}$ for i satisfying $\ s_0+1\leq i\leq m$ are sub-fractional since we have arranged the functions so that their growth rates are in descending order.

Once again, we rewrite the sequence $v(n)$ in the form

$$ \begin{align*} v(n)= \prod_{i=1}^{s_0} u_i^{p_i(n)+x_i(n)} \prod_{i=s_0+1}^{m} u_i^{p_i(n)+x_i(n)} \ \prod_{i=1}^{\ell}\ w_i^{\widetilde{p}_i(n)+y_i(n)}. \end{align*} $$

Because the function F is continuous, we can discard the functions $y_1,\ldots ,y_{\ell }$ , since they all converge to zero. The hypotheses in equations (8) and (9) on the linear combinations of the remaining functions in the exponents continue to hold. Indeed, this can be seen by noting that equations (8) and (9) still hold when replacing one of the functions (say $a_1$ ) by a function of the form $a_1(t)+e(t)$ , with $e(t)\to 0$ . Consequently, we can redefine $v(n)$ to be the sequence

$$ \begin{align*} v(n)= \prod_{i=1}^{s_0} u_i^{p_i(n)+x_i(n)} \prod_{i=s_0+1}^{m} u_i^{p_i(n)+x_i(n)} \ \prod_{i=1}^{\ell}\ w_i^{\widetilde{p}_i(n)}. \end{align*} $$

We will now reduce our problem to the case that the polynomials $\widetilde {p}_1(t),\ldots ,\widetilde {p}_{\ell }(t)$ are linearly independent. Due to our hypothesis (namely equations (8),(9)), every non-trivial linear combination of the functions $\widetilde {p}_1(t),\ldots ,\widetilde {p}_{\ell }(t)$ must satisfy either equation (8) or equation (9). Thus, every linear combination of the polynomials $\widetilde {p}_1(t),\ldots ,\widetilde {p}_{\ell }(t)$ is not a polynomial with integer coefficients unless it is the zero polynomial. If the second case is true, there exist $c_1,\ldots ,c_{\ell -1}\in \mathbb {R}$ such that

$$ \begin{align*} \widetilde{p}_{\ell}=c_1\widetilde{p}_1+\cdots+ c_{\ell -1}\widetilde{p}_{\ell-1}. \end{align*} $$

Then, we have

$$ \begin{align*}\prod_{i=1}^{\ell} w_{i}^{\widetilde{p}_i(n)}=\prod_{i=1}^{\ell-1}(w_{i}w_{\ell}^{c_i})^{\widetilde{p}_i(n)}. \end{align*} $$

If the polynomials $\widetilde {p}_1,\ldots ,\widetilde {p}_{\ell -1}$ are linearly independent, then we are done. Otherwise, we proceed similarly to eliminate $\widetilde {p}_{\ell -1}$ . After a finite number of steps, we will reach a collection of linearly independent polynomials.

In view of the above, we are allowed to assume that $\widetilde {p}_1,\ldots ,\widetilde {p}_{\ell }$ are linearly independent. Now, we show that we can reduce to the case that the polynomials $p_{s_0+1},\ldots , p_{m},\widetilde {p}_{1},\ldots , \widetilde {p}_{\ell }$ . Indeed, the linear independence assumption on the polynomials $\widetilde {p}_1,\ldots , \widetilde {p}_{\ell }$ implies that the polynomials $p_{s_0+1},\ldots ,p_m,\widetilde {p}_1,\ldots ,\widetilde {p}_{\ell }$ are linearly independent. To see how this works, observe that if there are real numbers $c_i,d_j$ such that

$$ \begin{align*}\sum_{i=1}^{m-s_0}c_ip_{s_0+i}+\sum_{i=1}^{\ell}d_ip^{\prime}_i=0,\end{align*} $$

then the function

$$ \begin{align*} \sum_{i=1}^{m-s_0}c_i(p_{s_0+i}+x_{s_0+i})+\sum_{i=1}^{\ell}d_i\widetilde{p}_i= \sum_{i=1}^{m-s_0}c_ix_{s_0+i} \end{align*} $$

is a sub-fractional function that does not converge to 0, since the functions $x_{s_0+i}$ are sub-fractional and have pairwise distinct growth rates. This contradicts our hypothesis (specifically equation (8)) and our claim follows.

In conclusion, we see that the sequence $v(n)$ can be written in the form

(23) $$ \begin{align} \prod_{i=1}^{s_0} u_i^{p_i(n)+x_i(n)} \prod_{i=s_0+1}^{m}u_i^{p_i(n)+x_i(n)} \prod_{i=1}^{\ell}w_i^{\widetilde{p}_i(n)}, \end{align} $$

where the functions $x_{i}$ are strongly non-polynomial with distinct growth rates, the functions $x_1,\ldots ,x_{s_0}$ are not sub-fractional, the functions $x_{s_0+1},\ldots ,x_s$ are sub-fractional and every non-trivial linear combination of the polynomials $p_{s_0+1},\ldots ,p_m,\widetilde {p}_1,\ldots ,\widetilde {p}_{\ell }$ is not an integer polynomial. We also recall that we have arranged the functions $x_i$ to be in decreasing order with respect to their growth rates.

We can combine the last two factors of this product into one factor to simplify our problem a bit more. More specifically, we can rewrite the sequence $v(n)$ in the form (we make some mild modifications in our notation here)

(24) $$ \begin{align} v(n)=\prod_{i=1}^{s_0} u_i^{p_i(n)+x_i(n)} \prod_{i=s_0+1}^{s}u_i^{\widetilde{p}_i(n)+x_i(n)}, \end{align} $$

where $s=m+l$ , $p_i,\widetilde {p}_j$ are real polynomials, the functions $x_{i}$ are strongly non-polynomial with distinct growth rates, $x_1,\ldots ,x_{s_0}$ are not sub-fractional, $x_{s_0+1},\ldots ,x_s$ are sub-fractional and every non-trivial linear combination of the polynomials $\widetilde {p}_{i}$ is not an integer polynomial. Namely, our functions satisfy hypotheses (a) and (b) of Lemma 4.2.

To establish our assertion, it suffices to show that the sequence $v(n)x$ (where $v(n)$ is as in equation (24)) is equidistributed on the nilmanifold $X'=\overline {u_1^{\mathbb {R}}\cdots u_s^{\mathbb {R}}x}$ for any $x\in X$ . We will prove this in the case $x=\Gamma $ since the general case follows from this using the change of base point trick that we discuss in Appendix B (see §B.1.2). In addition, we can invoke Lemma B.2 to find a real number $s_0$ , such that $X'=\overline {(u_1^{s_0})^{\mathbb {Z}}\ldots (u_s^{s_0})^{\mathbb {Z}}\Gamma }$ . Replacing the functions $p_i(t)+x_i(t)$ ( $1\leq i\leq s_0$ ) by the functions $(p_i(t)+x_i(t))/s_0$ and $\widetilde {p}_i(t)+x_i(t)$ ( $s_0+1\leq i\leq s$ ) by $(\widetilde {p}_i(t)+x_i(t))/s_0$ (the assumptions on the linear combinations of the functions remain unaffected), we can reduce our problem to the case that $X'=\overline {u_1^{\mathbb {Z}}\ldots u_s^{\mathbb {Z}}\Gamma }$ .

We want to show that for any continuous function F from $X'=G'/\Gamma '$ ( $G'$ is connected, simply connected and $\Gamma '$ is a uniform subgroup), the averages

$$ \begin{align*} \underset{1\leq n\leq N}{\operatorname*{\mathrm{\mathbb{E}}}} F(v(n)\Gamma') \end{align*} $$

converge to the integral $\int _{X'}F \ dm_{X'}$ . Since Lipschitz functions are dense in the space $C(X')$ , we may assume that F is Lipschitz continuous. In addition, we may assume after rescaling that $\lVert F \rVert _{\text {Lip}(X')}\leq 1$ . Now, our claim follows immediately from Lemma 4.2.

Proof of Lemma 4.2

Step 1. Approximating by polynomials. Let $L(t)$ be a sub-linear function with $\lim \nolimits _{t\to +\infty }L(t)=+\infty $ that we will determine later. It suffices to show that the sequence of the averages

(25) $$ \begin{align} \underset{N\leq n\leq N+L(N)}{\operatorname*{\mathrm{\mathbb{E}}}} F(v(n)\Gamma') \end{align} $$

converges to $\int _{X'}F \ d m_{X'}$ , since the conclusion would follow from Lemma 3.1. Reordering if necessary, we assume again that

$$ \begin{align*} x_1(t)\succ\cdots \succ x_{s_0}(t). \end{align*} $$

Let r be a very large natural number compared to the degrees of the polynomials $p_i,\widetilde {p}_j$ and the degrees of the functions $x_i(t)$ . If r is sufficiently large, we have that $x_i^{(r)}(t)=o_t(1)$ for all $i\in \{1,\ldots ,s_0\}$ . Assuming again that r is sufficiently large, then for any function $L(t)$ that satisfies

$$ \begin{align*} (x_i^{(r)}(t))^{-1/r} \prec L(t)\prec t^{1-\varepsilon'} \end{align*} $$

for some $\varepsilon '>0$ and all $i\in {1,\ldots ,s_0}$ , we have that for each $ i\in \{1,\ldots ,s_0\}$ , there is a unique natural number $k_i\geq r$ so that the sub-class $S(x_i,k_i)$ contains the function $L(t)$ (this follows from Proposition A.3). The fact that the function $L(t)$ belongs to $S(x_i,k_i)$ indicates that we have the relations

(26) $$ \begin{align} (x_i^{(k_i)}(t))^{-1/k_i} \prec L(t)\prec (x_i^{(k_i+1)}(t))^{-1/(k_i+1)}. \end{align} $$

We can guarantee that the numbers $k_i$ are also very large compared to the degrees of the polynomials $p_{j},\widetilde {p}_{j'}$ by enlarging the number r in the beginning. (For example, assuming that $k_i$ is at least 10 times as large as the maximal degree appearing among the polynomials $p_i,\widetilde {p}_j$ and 10 times as large as the number s of all existing polynomials would suffice for our arguments.)

We use the Taylor expansion for the functions $x_1(t),\ldots ,x_{s_0}(t)$ to write

(27) $$ \begin{align} x_i(N+h)=x_i(N)+\cdots +\frac{x_{i}^{k_i}(N)h^k}{k_i !}+o_N(1)= q_{i,N}(h)+o_N(1) \end{align} $$

for $0\leq h\leq L(N)$ (for the explanation of the $o_N(1)$ term, see the discussion after Proposition A.2). If, however, we have $i>s_0$ (namely, in the case where the function $x_i$ is sub-fractional), then

(28) $$ \begin{align} \max_{0\leq h\leq L(N)} |x_i(N+h)-x_i(N)|=o_N(1). \end{align} $$

In addition, we denote $p_{i,N}(h)=p_i(N+h)$ and similarly $\widetilde {p}_{i,N}(h)=\widetilde {p}_i(N+h)$ for every admissible value of i. Thus, we rewrite the expression in equation (25) as

(29) $$ \begin{align} \underset{0\leq h\leq L(N)}{\operatorname*{\mathrm{\mathbb{E}}}} F(w_N \prod_{i=1}^{s_0} u_i^{q_{i,N}(h)+p_{i,N}(h)} \prod_{i=s_0+1}^{s} u_i^{\widetilde{p}_{i,N}(h)} \Gamma'), \end{align} $$

where we discarded the $o_N(1)$ terms, because F is continuous. Here, $w_N=\prod _{i=s_0+1}^{s}u_i^{x_i(N)}$ but the explicit form of this term will not concern us, since we will only require that the element $w_N$ belongs to the underlying group $G'$ defining the nilmanifold $X'=\overline {u_1^{\mathbb {R}}\cdots u_s^{\mathbb {R}}\Gamma }$ .

In conclusion, we have reduced our problem to showing that given the nilmanifold $X'=\overline {u_1^{\mathbb {R}}\cdots u_s^{\mathbb {R}}\Gamma }$ (which is also equal to $\overline {u_1^{\mathbb {Z}}\cdots u_s^{\mathbb {Z}}\Gamma }$ ), the averages in equation (29) converge. Here, the polynomials $q_{i, N}$ are defined in equation (27) (they are essentially the Taylor polynomials of the Hardy field functions $x_i$ ), while the polynomials $p_{i, N},\widetilde {p}_{j, N}$ were defined by the relations $p_{i, N}=p_i(N+h) \text { and }\ \widetilde {p}_{j, N}=\widetilde {p}_j(N+h)$ , where the $p_i,\widetilde {p}_j$ are polynomials with real coefficients. We also recall that the polynomials $\widetilde {p}_i$ are such that every non-trivial linear combination of them is not an integer polynomial. Under all these assumptions, we will show that the polynomial sequence (restricted to the range $0\leq h\leq L(N)$ ) inside the function F is $\delta $ -equidistributed for N sufficiently large in the following step. We remark that the growth conditions in equation (26) imposed on the function $L(t)$ will also play a crucial role in this.

Step 2. Using the quantitative equidistribution theorem. Let $Z\cong \mathbb {T}^{d}$ be the horizontal torus of the nilmanifold $X'=\overline {u_1^{\mathbb {R}}\cdots u_s^{\mathbb {R}}\Gamma }$ and let $\pi :X'\to Z$ denote the projection map. Let $\delta>0$ be sufficiently small (in the sense that Theorem F is applicable). We assert that the finite polynomial sequence

(30) $$ \begin{align} \bigg( \prod_{i=1}^{s_0} u_i^{q_{i,N}(h)+p_{i,N}(h)} \prod_{i=s_0+1}^{s} u_i^{\widetilde{p}_{i,N}(h)} \Gamma' \bigg)_{0\leq h\leq L(N)} \end{align} $$

is $\delta $ -equidistributed on the nilmanifold $X'$ for N sufficiently large. If the claim does not hold for a natural number N, then by Theorem F, there exists a real number $M>0$ and a non-trivial horizontal character $\chi _N$ of modulus $\leq M$ such that

(31) $$ \begin{align} \bigg\lVert \chi_N\circ \pi \bigg( \prod_{i=1}^{s_0} u_i^{q_{i,N}(h)+p_{i,N}(h)} \prod_{i=s_0+1}^{s} u_i^{\widetilde{p}_{i,N}(h)} \Gamma'\bigg) \bigg\rVert_{C^{\infty}[L(N)]}\leq M. \end{align} $$

(The constant M depends only on $\delta $ , the nilmanifold $X'$ as well as the degrees of the polynomials $q_i,p_i$ , which are all fixed in our arguments. The central property we need is that it is independent of the variable N.) Thus, if our prior assertion fails, then the above relation would hold for infinitely many $N\in \mathbb {N}$ .

Our first goal is to eliminate the dependence of the characters $\chi _N$ on the variable N. Note that the function $\chi _N\circ \pi $ is a character on $\mathbb {T}^d$ of modulus $\leq M$ and, thus, has the form

$$ \begin{align*}(t_1,\ldots,t_{d})\to e(k_{1,N}t_1+\cdots+k_{d,N} t_d)\end{align*} $$

for $k_{i,N}\in \mathbb {Z} $ with $|k_{1,N}|+\cdots +|k_{d,N}|\leq M$ . We also write $\pi (u_i)= (u_{i,1},\ldots ,u_{i,d})$ for the projections of the elements $u_i$ on the horizontal torus. Then, a straightforward computation allows us to rewrite equation (31) as

(32) $$ \begin{align} &\bigg\lVert e\bigg(\sum_{i=1}^{s_0}(q_{i,N}(h)+p_{i,N}(h))(k_{1,N}u_{i,1}+\cdots+ k_{d,N}u_{i,d})\nonumber\\ &\quad+\sum_{i=s_0+1}^{s}(\widetilde{p}_{i,N}(h))(k_{1,N}u_{i,1}+\cdots+ k_{d,N}u_{i,d})\bigg) \bigg\rVert_{C^{\infty}[L(N)]}\leq M. \end{align} $$

Since there are only finitely many choices for the numbers $k_{1, N},\ldots ,k_{d, N}$ , we have that if our claim fails, there are $k_1,\ldots ,k_d\in \mathbb {Z}$ , so that the inequality

(33) $$ \begin{align} &\bigg\lVert e\bigg(\sum_{i=1}^{s_0}(q_{i,N}(h)+p_{i,N}(h))(k_{1}u_{i,1}+\cdots+ k_{d}u_{i,d})\nonumber\\ &\quad+\sum_{i=s_0+1}^{s}(\widetilde{p}_{i,N}(h))(k_{1}u_{i,1}+\cdots+ k_{d}u_{i,d})\bigg) \bigg\rVert_{C^{\infty}[L(N)]}\leq M \end{align} $$

holds for infinitely many $N\in \mathbb {N}$ . We will also denote the horizontal character corresponding to the d-tuplet $(k_1,\ldots ,k_d)$ by $\chi $ . Thus, we have eliminated the dependence of the character $\chi $ on N.

Denote $\widetilde {u_i}=k_1u_{i,1}+\cdots + k_du_{i,d}$ . We will show that the above hypotheses imply that all the numbers $\widetilde {u_i}$ equal $0$ . Thus, suppose that this is not valid and we will reach a contradiction. We consider two cases.

Case 1. First, suppose that all of the numbers $\widetilde {u_i}$ with $1\leq i\leq s_0$ are zero, which implies that the first summand in equation (33) vanishes. Naturally, equation (33) is simplified to

(34) $$ \begin{align} \bigg\lVert e\bigg(\sum_{i=s_0+1}^{s}\widetilde{p}_{i,N}(h)\widetilde{u_i}\bigg) \bigg\rVert_{C^{\infty}[L(N)]}\leq M. \end{align} $$

We recall here that we had defined $\widetilde {p}_{i,N}(h)=\widetilde {p}_i(N+h)$ . Let $Q(t)= \sum _{i=s_0+1}^{s}\widetilde {u_i}\widetilde {p}_{i}(t)$ . This is a linear combination of the polynomials $\widetilde {p}_i(t)$ . However, this linear combination is not a polynomial in $\mathbb {Q}[t]$ due to our assumptions on the polynomials $\widetilde {p}_i(n)$ , unless, of course, all the coefficients $\widetilde {u_i}$ (for $s_0+1\leq i\leq s$ ) in this combination are zero, which we have supposed to not be the case. Thus, $Q(t)$ has at least one irrational coefficient (except the constant term) and is equidistributed on $\mathbb {T}$ . The relation in equation (34) implies that $\lVert e(Q(N+h)) \rVert _{C^{\infty }[L(N)]}\leq M$ for infinitely many N. It is not difficult to see by calculating the coefficients in $Q(N+h)$ that this fails for N large enough.

Case 2. Suppose now that at least one of the numbers $\widetilde {u_i}$ with $1\leq i\leq s_0$ is non-zero. Furthermore, assume l is a positive integer that is larger than the degrees of the polynomials $p_{i, N}(h),\widetilde {p}_{j, N}(h)$ (for all admissible values of the indices $i, j$ ) as well as the degrees of the functions $x_i$ , but l is also smaller than all the numbers $k_i$ . Recall that we have picked $k_i$ to be very large in relation to the degrees of the polynomials $p_{i},\widetilde {p}_{j}$ and degrees of the functions $x_i$ in the beginning, thus we can find ‘many’ such numbers l. The fact that l is larger than the degrees of the functions $x_i$ combined with Proposition A.1 implies that $x_i^{(l)}(t)\to 0$ , as $t\to +\infty $ .

For a number l as above, the coefficient of the term $h^{l}$ in the polynomial appearing in equation (33) is equal to

$$ \begin{align*} \frac{1}{l!}\sum_{i=1}^{s_0}x_i^{(l)}(N) \widetilde{u_i} \end{align*} $$

and, thus, it does not depend on the polynomials $p_i,\widetilde {p}_j$ . Using the definition of the smoothness norms, equation (33) implies that

$$ \begin{align*} L(N)^{l}\bigg\lVert \frac{1}{l!}\sum_{i=1}^{s_0}x_i^{(l)}(N) \widetilde{u_i} \bigg\rVert_{\mathbb{T}}\leq M \end{align*} $$

for infinitely many $N\in \mathbb {N}$ . The last inequality becomes

$$ \begin{align*} L(N)^{l}\bigg|\sum_{i=1}^{s_0}x_i^{(l)}(N)\widetilde{u_i}\bigg|\leq l!M \end{align*} $$

for large enough N, because all functions $x_i^{l}(t)$ go to 0. However, the Hardy field function inside the absolute value above has the same growth rate as the function $x_1^{(l)}(t)$ , since the functions $x_1,\ldots ,x_{s_0}$ are strongly non-polynomial and have distinct growth rates (recall that $x_1$ has the largest growth rate among the $x_i$ ), unless, of course, $\widetilde {u}_1=0$ . If the latter does not hold, we get

$$ \begin{align*} \bigg|x_1^{(l)}(N)\widetilde{u}_1 \bigg|\leq \frac{C}{L(N)^l} \end{align*} $$

for infinitely many N and some constant C, which contradicts equation (26). Thus, we eventually deduce that $\widetilde {u}_1=k_1u_{1,1}+\cdots + k_du_{1,d}=0$ . Repeating the same argument, we get inductively that $\widetilde {u}_i=k_1u_{i,1}+\cdots + k_du_{i,d}$ =0 for all $1\leq i\leq s_0$ , which is a contradiction.

To summarize, we have shown that if the sequence in equation (30) is not $\delta $ -equidistributed for all large enough N, then all the numbers $\widetilde {u}_i=k_1u_{i,1}+\cdots + k_du_{i,d}$ are zero. Equivalently, we have $\chi \circ \pi (u_i)=0$ for all $1\leq i\leq s$ . This implies that the character $\chi $ is the trivial character on $X'$ . Indeed, the character $\chi $ annihilates all elements $u_1^{n_1}\cdots u_s^{n_s}\Gamma $ , where $n_1,\ldots ,n_s\in \mathbb {Z}$ and by density of those elements on $X'$ (recall our assumption that $X'$ is also equal to the nilmanifold $\overline {u_1^{\mathbb {Z}}\cdots u_s^{\mathbb {Z}}\Gamma }$ ), $\chi $ is zero everywhere. This is a contradiction (the horizontal characters appearing when we applied Theorem F are assumed to be non-trivial).

In conclusion, we have that the finite polynomial sequence in equation (30) is $\delta $ -equidistributed for N sufficiently large. Thus, we conclude that the averages in equation (25) are $\delta \lVert F(w_N \cdot ) \rVert _{\text {Lip}(X')}=\delta \lVert F \rVert _{\text {Lip}(X')}$ close to the quantity $ \int _{X'} F(w_N x)\ dm_{X'}(x)$ . The action of $w_N$ on $X'$ preserves the Haar measure of $X'$ , so we get that the last integral is equal to $\int _{X'} F(x) \ dm_{X'}(x)$ . Taking $\delta \to 0$ , we finish the proof.

Proof of Theorem 1.1

As we explained in the previous section (before the statement of Lemma 3.2), the first part follows from the second part (see also [Reference Frantzikinakis4, Lemma 5.1]) and, in turn, this second part follows using similar arguments as in the proof of Theorem 1.2. We only highlight the main differences here. All the disparities appear in the part where we reduce Proposition 4.1 to Lemma 4.2.

(a) In equation (22), all the functions $f_1,\ldots ,f_s$ satisfy equation (8) (there are no functions among the $f_i$ that satisfy $\lim \nolimits _{t\to +\infty }|f_i(t)|<\infty $ ). We also have $k=s$ .

(b) We do not have to make the reduction to the case where the polynomials $\widetilde {p}_1,\ldots ,\widetilde {p}_{\ell }$ are linearly independent. There cannot be a non-trivial linear combination of them that is zero, because that would violate equation (8).

(c) The limit of the averages is again $\int _{X'} F(x) \ dm_{X'}(x)$ , where $X'=\overline {u_1^{\mathbb {R}}\cdots u_s^{\mathbb {R}}\Gamma }$ by Lemma 4.2. We would like to show that the limit is equal to $\int _{X"} Fdm_X"$ , where $X"$ is the nilmanifold $\overline {b_1^{\mathbb {R}}\cdots b_k^{\mathbb {R}}\Gamma }$ . Recall that each $u_i$ is equal to $b_1^{c_{i,1}}\cdots b_k^{c_{i,k}}$ (by equation (21)) and the numbers $c_{i,j}$ form an invertible $k\times k$ matrix (due to the linear independence assumption on the original functions $a_1,\ldots ,a_k$ ). Thus, we can also write $b_i=\prod _{j=1}^{k}u_i^{c^{\prime }_{i,j}}$ for some numbers $c^{\prime }_{i,j}$ (here, we also use that the elements $b_i$ are pairwise commuting). Combining the above, we have that $b_1^{\mathbb {R}}\cdots b_k^{\mathbb {R}}=u_1^{\mathbb {R}}\cdots u_k^{\mathbb {R}}$ and thus the closures of their projections on $G/\Gamma $ define the same subnilmanifold.