1. Mahler’s and Koksma’s classifications of p-adic numbers

Let p be a prime number and let $|\cdot |_{p}$ denote the p-adic absolute value on the field $\mathbb {Q}$ of rational numbers, normalised such that $|p|_{p}=p^{-1}$ . The completion of $\mathbb {Q}$ with respect to $|\cdot |_{p}$ is the field $\mathbb {Q}_{p}$ of p-adic numbers, and the unique extension of $|\cdot |_{p}$ to the field $\mathbb {Q}_{p}$ is denoted by the same notation $|\cdot |_{p}$ . Mahler [Reference Mahler16] gave a classification of p-adic numbers in analogy with his classification [Reference Mahler15] of real numbers, as follows. Let $P(x)=a_{n}x^{n}+\cdots +a_{1}x+a_{0}$ be a nonzero polynomial in x over the ring $\mathbb {Z}$ of rational integers. We denote by $\deg (P)$ the degree of $P(x)$ with respect to x. The height $H(P)$ of $P(x)$ is defined by $H(P)=\max \{|a_{n}|,\ldots ,|a_{1}|, |a_{0}|\}$ , where $|\cdot |$ denotes the usual absolute value on the field $\mathbb {R}$ of real numbers. Let $\xi $ be any p-adic number and let n, H be any positive rational integers. Following Bugeaud [Reference Bugeaud3], set

$$ \begin{align*} w_{n}(H, \xi)=\min\{|P(\xi)|_{p}: P(x)\in \mathbb{Z}[x], \hspace{0.03in} \deg(P)\leq n, H(P)\leq H \textrm{ and } P(\xi)\neq0\}, \end{align*} $$

$$ \begin{align*} w_{n}(\xi)=\limsup_{H \rightarrow \infty}\frac{-\log (H w_{n}(H, \xi))}{\log H} \quad\textrm{and}\quad w(\xi)=\limsup_{n \rightarrow \infty}\frac{w_{n}(\xi)}{n}. \end{align*} $$

Then $\xi $ is called:

• • a p-adic A-number if $w(\xi )=0$ ;

• • a p-adic S-number if $0< w(\xi )<\infty $ ;

• • a p-adic T-number if $w(\xi )=\infty $ and $w_{n}(\xi )<\infty $ for $n=1,2,3,\ldots $ ; and

• • a p-adic U-number if $w(\xi )=\infty $ and $w_{n}(\xi )=\infty $ from some n onward.

The set of p-adic A-numbers coincides with the set of algebraic p-adic numbers. Therefore, the transcendental p-adic numbers are separated into the three disjoint classes S, T and U. If $\xi $ is a p-adic U-number and m is the minimum of the positive integers n satisfying $w_{n}(\xi )=\infty $ , then $\xi $ is called a p-adic $U_{m}$ -number. Alnıaçık [Reference Alnıaçık1, Ch. III, Theorem I] gave the first explicit constructions of p-adic $U_{m}$ -numbers for each positive integer m. For further constructions of p-adic S-, T- and U-numbers, see [Reference Bugeaud and Kekeç4, Reference Bugeaud and Kekeç5, Reference Kekeç9, Reference Kekeç10].

Assume that $\alpha $ is an algebraic p-adic number. Let $P(x)$ be the minimal polynomial of $\alpha $ over $\mathbb {Z}$ . Then the degree $\deg (\alpha )$ of $\alpha $ and the height $H(\alpha )$ of $\alpha $ are defined by ${\deg (\alpha )=\deg (P)}$ and $H(\alpha )=H(P)$ . Given a p-adic number $\xi $ and positive rational integers n, H, in analogy with Koksma’s classification [Reference Koksma12] of real numbers and as in Bugeaud [Reference Bugeaud3] and Schlickewei [Reference Schlickewei21]), set

$$ \begin{align*} w_{n}^{*}(H, \xi)=\min\left\{|\xi-\alpha|_{p}:\! \begin{array}{ll} \alpha\textrm{ is an algebraic }p\textrm{-adic number},\\ \deg(\alpha)\leq n, H(\alpha)\leq H \textrm{ and } \alpha\neq\xi \end{array}\!\!\right\}, \end{align*} $$

$$ \begin{align*} w_{n}^{*}(\xi)=\limsup_{H \rightarrow \infty}\frac{-\log (H w_{n}^{*}(H, \xi))}{\log H} \quad \textrm{and} \quad w^{*}(\xi)=\limsup_{n \rightarrow \infty}\frac{w_{n}^{*}(\xi)}{n}. \end{align*} $$

Then $\xi $ is called:

• • a p-adic $A^{*}$ -number if $w^{*}(\xi )=0$ ;

• • a p-adic $S^{*}$ -number if $0< w^{*}(\xi )<\infty $ ;

• • a p-adic $T^{*}$ -number if $w^{*}(\xi )=\infty $ and $w_{n}^{*}(\xi )<\infty $ for $n=1,2,3,\ldots $ ; and

• • a p-adic $U^{*}$ -number if $w^{*}(\xi )=\infty $ and $w_{n}^{*}(\xi )=\infty $ from some n onward.

The set of p-adic $A^{*}$ -numbers is equal to the set of algebraic p-adic numbers. Therefore, the transcendental p-adic numbers are separated into the three disjoint classes $S^{*}$ , $T^{*}$ and $U^{*}$ . Let $\xi $ be a p-adic $U^{*}$ -number and let m be the minimum of the positive integers n satisfying $w_{n}^{*}(\xi )=\infty $ . Then $\xi $ is called a p-adic $U_{m}^{*}$ -number. Mahler’s classification of p-adic numbers is equivalent to Koksma’s classification of p-adic numbers, that is, the classes A, S, T and U are the same as the classes $A^{*}$ , $S^{*}$ , $T^{*}$ and $U^{*}$ , respectively. Furthermore, a p-adic $U_{m}^{*}$ -number is a p-adic $U_{m}$ -number and vice versa. (See Bugeaud [Reference Bugeaud3] for further information on Mahler’s and Koksma’s classifications of p-adic numbers.)

2. Ruban p-adic continued fractions

Ruban [Reference Ruban20] introduced a continued fraction algorithm in $\mathbb {Q}_{p}$ . In this section, we recall the Ruban p-adic continued fraction algorithm and its basic properties following the approach of Perron [Reference Perron19, Sections 29 and 30, pages 101–108] (see also [Reference Laohakosol14, Reference Ooto17, Reference Wang22, Reference Wang23]). Let $\xi $ be a nonzero p-adic number with the canonical expansion

$$ \begin{align*} \xi=\sum_{j=k}^{\infty}a_{j}p^{j}, \end{align*} $$

where $a_{j}\in \{0,1,\ldots ,p-1\}$ for $j=k,k+1,\ldots , a_{k}\neq 0$ and k is the rational integer such that $|\xi |_{p}=p^{-k}$ . If $k\leq 0$ , then we write $\xi =\{\xi \}+\lfloor \xi \rfloor $ , where

$$ \begin{align*} \{\xi\}=\sum_{j=k}^{0}a_{j}p^{j} \quad\textrm{and}\quad \lfloor\xi\rfloor=\sum_{j=1}^{\infty}a_{j}p^{j}. \end{align*} $$

If $k>0$ , then we write $\xi =\{\xi \}+\lfloor \xi \rfloor $ , where

$$ \begin{align*} \{\xi\}=0 \quad \textrm{and}\quad \lfloor\xi\rfloor=\sum_{j=k}^{\infty}a_{j}p^{j}. \end{align*} $$

Further, we write $0=\{0\}+\lfloor 0\rfloor $ , where $\{0\}=\lfloor 0\rfloor =0$ . Then, for each p-adic number $\xi $ , $\{\xi \}$ and $\lfloor \xi \rfloor $ are uniquely determined. Let $b_{0}, b_{1}, b_{2},\ldots $ be nonnegative rational numbers with

$$ \begin{align*} b_{0}\in\{\{\xi\}: \xi\in \mathbb{Q}_{p}\} \quad \textrm{and} \quad b_{\nu}\in\{\{\xi\}: \xi\in \mathbb{Q}_{p}, \ |\xi|_{p}\geq p\} \quad (\nu=1,2,3,\ldots). \end{align*} $$

A finite Ruban p-adic continued fraction $[b_{0}, b_{1},\ldots ,b_{n}]_{p}$ is defined by

$$ \begin{align*} [b_0, b_1, \ldots, b_{n}]_{p}=b_0+\dfrac{1}{b_1+\dfrac{1}{\begin{array}{lr}\ddots\\ \qquad +\dfrac{1}{b_n}\end{array}}}. \end{align*} $$

Then we have the following properties.

$$ \begin{align*} [b_{0}]_{p}=b_{0}, \quad [b_{0}, b_{1}]_{p}=b_{0}+\frac{1}{b_{1}}, \end{align*} $$

$$ \begin{align*} [b_0, b_1, \ldots, b_{n}]_{p}=\bigg[b_0, b_1, \ldots, b_{n-2},b_{n-1}+\frac{1}{b_{n}}\bigg]_{p}=[b_0, b_1, \ldots, b_{m-1},[b_{m},\ldots,b_{n}]_{p}]_{p}, \end{align*} $$

$$ \begin{align*} [b_0, b_1, \ldots, b_{n}]_{p}=b_{0}+\frac{1}{[b_1, \ldots, b_{n}]_{p}}. \end{align*} $$

Hence, $[b_0, b_1, \ldots , b_{n}]_{p}$ is a nonnegative rational number, and the numbers ${b_{\nu}\ (\nu =0,1,\ldots ,n)}$ are called the partial quotients of the Ruban p-adic continued fraction $[b_0, b_1, \ldots , b_{n}]_{p}$ . Define the nonnegative rational numbers $p_{\nu }$ and $q_{\nu }$ by

(2.1) $$ \begin{align} \begin{cases}p_{-2}=0, \quad p_{-1}=1, \quad p_{\nu}=b_{\nu}p_{\nu-1}+p_{\nu-2} & (\nu=0,1,2,\ldots), \\ q_{-2}=1, \quad q_{-1}=0, \quad q_{\nu}=b_{\nu}q_{\nu-1}+q_{\nu-2} &(\nu=0,1,2,\ldots). \end{cases} \end{align} $$

By induction,

$$ \begin{align*} [b_0, b_1, \ldots, b_{n}]_{p}=\frac{p_{n}}{q_{n}} \quad (n=0,1,2,\ldots). \end{align*} $$

The nonnegative rational numbers $p_{0}/q_{0}, p_{1}/q_{1},\ldots ,p_{n}/q_{n}$ are called the convergents of the Ruban p-adic continued fraction $[b_0, b_1, \ldots , b_{n}]_{p}$ ; $p_{\nu }/q_{\nu }\ (\nu =0,1,\ldots ,n)$ is called the $\nu $ th convergent of $[b_0, b_1, \ldots , b_{n}]_{p}$ . By induction,

(2.2) $$ \begin{align} p_{\nu}q_{\nu-1}-p_{\nu-1}q_{\nu}=(-1)^{\nu-1} \quad (\nu=-1,0,1,\ldots). \end{align} $$

From (2.1),

$$ \begin{align*} |q_{n}|_{p}=|b_{1}|_{p}\cdot|b_{2}|_{p}\cdots|b_{n}|_{p} \quad \textrm{and} \quad |p_{n}|_{p}=|b_{0}|_{p}\cdot|b_{1}|_{p}\cdots|b_{n}|_{p}=|b_{0}|_{p}\cdot|q_{n}|_{p} \ (\textrm{if } b_{0}\neq0) \end{align*} $$

for $n\kern1.4pt{=}\kern1.4pt 1,2,3,\ldots .$ As $|b_{\nu }|_{p}\kern1.4pt{\geq}\kern1.4pt p\ (\nu =1,2,3,\ldots )$ , we have $|q_{n+1}|_{p}\kern1.4pt{>}\kern1.4pt|q_{n}|_{p}$ and ${|p_{n+1}|_{p}\kern1.4pt{>}\kern1.4pt|p_{n}|_{p}}$ for $n=1,2,3,\ldots .$ Therefore,

$$ \begin{align*} \lim_{n\rightarrow\infty}|q_{n}|_{p}=\infty \quad \textrm{and} \quad \lim_{n\rightarrow\infty}|p_{n}|_{p}=\infty. \end{align*} $$

By (2.2),

$$ \begin{align*} \bigg|\frac{p_{n}}{q_{n}}-\frac{p_{n-1}}{q_{n-1}}\bigg|_{p}=\frac{1}{|q_{n}|_{p}\cdot|q_{n-1}|_{p}} \quad (n=1,2,3,\ldots). \end{align*} $$

Then

$$ \begin{align*} \lim_{n\rightarrow\infty}\bigg|\frac{p_{n}}{q_{n}}-\frac{p_{n-1}}{q_{n-1}}\bigg|_{p}=0. \end{align*} $$

Thus, $\{p_{n}/q_{n}\}_{n=0}^{\infty }$ is a Cauchy sequence in $\mathbb {Q}_{p}$ and has a limit in $\mathbb {Q}_{p}$ . An infinite Ruban p-adic continued fraction $[b_0, b_1, b_2, \ldots ]_{p}$ is defined as the limit of the sequence $\{p_{n}/q_{n}\}_{n=0}^{\infty }$ , that is,

$$ \begin{align*} [b_0, b_1, b_2, \ldots]_{p}:=\lim_{n\rightarrow\infty}\frac{p_{n}}{q_{n}}=\lim_{n\rightarrow\infty}[b_0, b_1, \ldots, b_{n}]_{p}. \end{align*} $$

Further, for $\xi \in \mathbb {Q}_{p}\setminus \{0\}$ ,

(2.3) $$ \begin{align} [b_0, \ldots, b_{n},\xi]_{p}=\frac{p_{n}\cdot\xi+p_{n-1}}{q_{n}\cdot\xi+q_{n-1}} \quad (n=0,1,2,\ldots). \end{align} $$

Let $\xi _{0}$ be a p-adic number. If $\xi _{0}\neq \{\xi _{0}\}$ , then we write

$$ \begin{align*} \xi_{0}=b_{0}+\frac{1}{\xi_{1}}, \end{align*} $$

where $b_{0}=\{\xi _{0}\}$ , $\xi _{1}=1/\lfloor \xi _{0}\rfloor $ , $|\xi _{1}|_{p}\geq p$ and $\{\xi _{1}\}\neq 0$ . If $\xi _{1}\neq \{\xi _{1}\}$ , then we write

$$ \begin{align*} \xi_{1}=b_{1}+\frac{1}{\xi_{2}}, \end{align*} $$

where $b_{1}=\{\xi _{1}\}$ , $\xi _{2}=1/\lfloor \xi _{1}\rfloor $ , $|\xi _{2}|_{p}\geq p$ and $\{\xi _{2}\}\neq 0$ . If the process continues, then

(2.4) $$ \begin{align} \xi_{\nu}=b_{\nu}+\frac{1}{\xi_{\nu+1}} \quad (\nu\geq0), \end{align} $$

where $b_{\nu }=\{\xi _{\nu }\}\ (\nu \geq 0)$ and $\xi _{\nu +1}=1/\lfloor \xi _{\nu }\rfloor\ (\nu \geq 0)$ , and

$$ \begin{align*} |\xi_{\nu}|_{p}=|b_{\nu}|_{p}\geq p \quad (\nu\geq1). \end{align*} $$

The p-adic numbers $\xi _{1}, \xi _{2},\ldots $ are called complete quotients, and the nonnegative rational numbers $b_{0}, b_{1}, b_{2},\ldots $ are called partial quotients. It follows from (2.4) that

(2.5) $$ \begin{align} \xi_{0}=[b_0, \xi_{1}]_{p}=[b_0, b_{1}, \xi_{2}]_{p}=[b_0, b_{1},\ldots,b_{n}, \xi_{n+1}]_{p} \end{align} $$

and

$$ \begin{align*} \xi_{\nu}=[b_\nu, b_{\nu+1},\ldots,b_{n}, \xi_{n+1}]_{p} \quad (\nu=0,1,\ldots,n). \end{align*} $$

By (2.5), (2.3) and (2.2),

$$ \begin{align*} \xi_{0}-\frac{p_{n}}{q_{n}}=\frac{p_{n}\xi_{n+1}+p_{n-1}}{q_{n}\xi_{n+1}+q_{n-1}}-\frac{p_{n}}{q_{n}}=\frac{(-1)^{n}}{q_{n}(q_{n}\xi_{n+1}+q_{n-1})}. \end{align*} $$

Then

(2.6) $$ \begin{align} \bigg|\xi_{0}-\frac{p_{n}}{q_{n}}\bigg|_{p}=\frac{1}{|\xi_{n+1}|_{p}\cdot|q_{n}|_{p}^{2}}=\frac{1}{|b_{n+1}|_{p}\cdot|q_{n}|_{p}^{2}}=\frac{1}{|q_{n+1}|_{p}\cdot|q_{n}|_{p}}<\frac{1}{|q_{n}|_{p}^{2}}. \end{align} $$

We now have two cases to consider.

Case (i). Some $\xi _{n+1}$ appears with $\xi _{n+1}=\{\xi _{n+1}\}=b_{n+1}$ and the process stops with $\xi _{n+1}=b_{n+1}$ . Then it follows from (2.5) that

$$ \begin{align*} \xi_{0}=[b_0, b_{1},\ldots,b_{n}, b_{n+1}]_{p}. \end{align*} $$

Case (ii). $\xi _{n+1}\neq \{\xi _{n+1}\}$ for every $n\geq -1$ and the process never stops. Then it follows from (2.6) that

$$ \begin{align*} \xi_{0}=\lim_{n\rightarrow\infty}\frac{p_{n}}{q_{n}}=\lim_{n\rightarrow\infty}[b_0, b_1, \ldots, b_{n}]_{p}=[b_0, b_1, b_2, \ldots]_{p}. \end{align*} $$

The Ruban continued fraction expansion of a p-adic number is unique because the canonical expansion of a p-adic number is unique. Laohakosol [Reference Laohakosol14] and Wang [Reference Wang22] proved that a p-adic number is rational if and only if its Ruban continued fraction expansion is finite or ultimately periodic with the period $p-p^{-1}$ . Ooto [Reference Ooto17] recently proved that an analogue of Lagrange’s theorem does not hold for the Ruban p-adic continued fraction: that is, there are quadratic irrational p-adic numbers whose Ruban continued fraction expansions are not ultimately periodic.

3. Our main results

Alnıaçık [Reference Alnıaçık2, Theorem] gave a construction of real $U_{m}$ -numbers by using continued fraction expansions of algebraic irrational real numbers of degree m. In the present paper, we establish the following p-adic analogue.

Theorem 3.1. Let $\alpha $ be an algebraic irrational p-adic number with $|\alpha |_{p}\geq 1$ and the Ruban p-adic continued fraction expansion

(3.1) $$ \begin{align} \alpha=[a_0,a_1,a_2,\ldots]_{p}. \end{align} $$

Let $(r_{n})_{n=0}^{\infty }$ and $(s_{n})_{n=0}^{\infty }$ be two infinite sequences of nonnegative rational integers such that

$$ \begin{align*} 0=r_{0}< s_{0}< r_{1}< s_{1}< r_{2}< s_{2}< r_{3}< s_{3}< \cdots \quad \textrm{and} \quad r_{n+1}-s_{n}\geq2. \end{align*} $$

Denote by $p_{n}/q_{n}\ (n=0,1,2,\ldots )$ the nth convergent of the Ruban p-adic continued fraction (3.1). Assume that

(3.2) $$ \begin{align} \lim_{n \rightarrow \infty}\frac{\log |q_{s_{n}}|_{p}}{\log |q_{r_{n}}|_{p}}=\infty \end{align} $$

and

(3.3) $$ \begin{align} \limsup_{n \rightarrow \infty}\frac{\log |q_{r_{n+1}}|_{p}}{\log |q_{s_{n}}|_{p}}<\infty. \end{align} $$

Define the rational numbers $b_{j}\, (j=0,1,2,\ldots )$ by

(3.4) $$ \begin{align} b_{j}=\begin{cases}a_{j} \quad \mbox{if }r_{n}\leq j\leq s_{n} & (n=0,1,2,\ldots), \\ \upsilon_{j} \quad \mbox{if }s_{n}<j<r_{n+1} & (n=0,1,2,\ldots), \end{cases} \end{align} $$

where $\upsilon _{j}$ is a rational number of the form

$$ \begin{align*} \upsilon_{j}=c_{-d}p^{-d}+c_{-d+1}p^{-d+1}+\cdots+c_{-1}p^{-1}+c_{0}. \end{align*} $$

Here, $d\in \mathbb {Z}$ , $d>0$ , $c_{-d}\neq 0$ and $c_{i}\in \{0,1,\ldots ,p-1\}$ for $i=-d, -d+1,\ldots , -1, 0$ . Note that $|\upsilon _{j}|_{p}\geq p$ . Suppose that $|\upsilon _{j}|_{p}\leq \kappa _{1}|a_{j}|_{p}^{\kappa _{2}}$ and $\sum _{j=s_{n}+1}^{r_{n+1}-1}|a_{j}-\upsilon _{j}|_{p}\neq 0$ , where $\kappa _{1}$ and $\kappa _{2}$ are fixed positive rational integers. Then the irrational p-adic number ${\xi =[b_0,b_1,b_2,\ldots ]_{p}}$ is a p-adic $U_{m}$ -number, where m denotes the degree of the algebraic irrational p-adic number $\alpha $ .

Recently, Kekeç [Reference Kekeç11, Theorem 1.5] modified the hypotheses in Alnıaçık [Reference Alnıaçık2, Theorem] and gave a construction of transcendental real numbers that are not U-numbers by using continued fraction expansions of irrational algebraic real numbers. Our second main result in the present paper is the following partial p-adic analogue of Kekeç [Reference Kekeç11, Theorem 1.5].

Theorem 3.3. Let $\alpha $ be an algebraic p-adic number of degree $m\geq 2$ with $|\alpha |_{p}\geq 1$ and the Ruban p-adic continued fraction expansion

$$ \begin{align*} \alpha=[a_0,a_1,a_2,\ldots]_{p}. \end{align*} $$

Let $(r_{n})_{n=0}^{\infty }$ and $(s_{n})_{n=0}^{\infty }$ be two infinite sequences of nonnegative rational integers such that

$$ \begin{align*} 0=r_{0}< s_{0}< r_{1}< s_{1}< r_{2}< s_{2}< r_{3}< s_{3}< \cdots \quad \textrm{and} \quad r_{n+1}-s_{n}\geq2. \end{align*} $$

Denote by $p_{n}/q_{n}\ (n=0,1,2,\ldots )$ the nth convergent of the Ruban p-adic continued fraction $\alpha $ . Define the rational numbers $b_{\,j}\ (j=0,1,2,\ldots )$ by

(3.5) $$ \begin{align} b_{j}= \begin{cases}a_{j} \quad \mbox{if }r_{n}\leq j\leq s_{n} & (n=0,1,2,\ldots), \\ \upsilon_{j} \quad \mbox{if }s_{n}<j<r_{n+1} & (n=0,1,2,\ldots), \end{cases} \end{align} $$

where $\upsilon _{j}$ is a rational number of the form

$$ \begin{align*} \upsilon_{j}=c_{-d}p^{-d}+c_{-d+1}p^{-d+1}+\cdots+c_{-1}p^{-1}+c_{0}. \end{align*} $$

Here $d\in \mathbb {Z}$ , $d>0$ , $c_{-d}\neq 0$ and $c_{i}\in \{0,1,\ldots ,p-1\}$ for $i=-d, -d+1,\ldots , -1, 0$ . Note that $|\upsilon _{j}|_{p}\geq p$ . Suppose that $|\upsilon _{j}|_{p}\leq \kappa _{1}|a_{j}|_{p}^{\kappa _{2}}$ and $\sum _{j=s_{n}+1}^{r_{n+1}-1}|a_{j}-\upsilon _{j}|_{p}\neq 0$ , where $\kappa _{1}$ and $\kappa _{2}$ are fixed positive rational integers. Assume that

(3.6) $$ \begin{align} \liminf_{n \rightarrow \infty}\frac{\log |q_{s_{n}}|_{p}}{\log |q_{r_{n}}|_{p}}>2+4m\bigg(m+\kappa_{2}+\frac{\log \kappa_{1}}{\log 2}\bigg). \end{align} $$

Then the irrational p-adic number $\xi =[b_0,b_1,b_2,\ldots ]_{p}$ is transcendental.

In the next section, we cite some auxiliary results that we need to prove our results. In Section 5, we prove Theorems 3.1 and 3.3.

4. Auxiliary results

The following lemma is a p-adic analogue of Alnıaçık [Reference Alnıaçık2, Lemma IV].

Lemma 4.1. Let $p/q$ and $u/v$ be two rational numbers with Ruban p-adic continued fraction expansions

$$ \begin{align*} \frac{p}{q}=[a_{0},a_{1},\ldots,a_{n}]_{p} \quad \textrm{and} \quad \frac{u}{v}=[b_{0},b_{1},\ldots,b_{n}]_{p} \quad (|a_{0}|_{p}\geq1, |b_{0}|_{p}\geq1). \end{align*} $$

Assume that

(4.1) $$ \begin{align} |b_{j}|_{p}\leq\kappa_{1}|a_{j}|_{p}^{\kappa_{2}}\quad (j=0,1,\ldots,n), \end{align} $$

where $\kappa _{1}$ and $\kappa _{2}$ are fixed positive rational integers. Then

$$ \begin{align*} |u|_{p}\leq |a_{0}|_{p}^{\kappa_{2}}\kappa_{1}|q|_{p}^{\kappa_{2}+\log \kappa_{1}{/}\kern-1pt\log 2}. \end{align*} $$

Proof. It follows from (4.1) that

$$ \begin{align*} |u|_{p}=|b_{0}|_{p}\cdot|b_{1}|_{p}\cdots|b_{n}|_{p}\leq \kappa_{1}^{n+1}\cdot(|a_{0}|_{p}\cdot|a_{1}|_{p}\cdots|a_{n}|_{p})^{\kappa_{2}}. \end{align*} $$

As $|q|_{p}=|a_{1}|_{p}\cdots |a_{n}|_{p}\geq p^{n}\geq 2^{n}$ ,

$$ \begin{align*} |u|_{p}\leq (2^{n+1})^{\log \kappa_{1}{/}\kern-1pt\log 2}|a_{0}|_{p}^{\kappa_{2}}|q|_{p}^{\kappa_{2}}\leq |a_{0}|_{p}^{\kappa_{2}}\kappa_{1}|q|_{p}^{\kappa_{2}+\log \kappa_{1}{/}\kern-1pt\log 2}.\\[-37pt] \end{align*} $$

Theorem 4.2 (Içen [Reference İçen8, page 25] and [Reference İçen7, Lemma 1, page 71]).

Let L be a p-adic algebraic number field of degree m and let $\alpha _1,\ldots ,\alpha _k$ be algebraic p-adic numbers in L. Let $\eta $ be any algebraic p-adic number. Suppose that $F(\eta ,\alpha _1,\ldots ,\alpha _k)=0$ , where $F(x,x_1,\ldots ,x_k)$ is a polynomial in $x, x_{1},\ldots ,x_{k}$ over $\mathbb {Z}$ with degree at least one in x. Then

$$\begin{align*}H(\eta)\leq 3^{2dm+(l_{1}+\cdots+l_{k})m}H^mH(\alpha_1)^{l_{1}m}\cdots H(\alpha_k)^{l_{k}m},\end{align*}$$

where d is the degree of $F(x,x_1,\ldots ,x_k)$ in x, $l_i$ is the degree of $F(x,x_1,\ldots ,x_k)$ in ${x_i\ (i=1,\ldots ,k)}$ and H is the maximum of the usual absolute values of the coefficients of $F(x,x_1,\ldots ,x_k)$ .

Lemma 4.3 (Pejkovic [Reference Pejkovic18, Lemma 2.5]).

Let $\alpha _{1}$ and $\alpha _{2}$ be two distinct algebraic p-adic numbers. Then

$$ \begin{align*} |\alpha_{1}-\alpha_{2}|_{p}\geq (\deg (\alpha_{1})+1)^{-\deg (\alpha_{2})}(\deg (\alpha_{2})+1)^{-\deg (\alpha_{1})}H(\alpha_{1})^{-\deg (\alpha_{2})}H(\alpha_{2})^{-\deg (\alpha_{1})}. \end{align*} $$

Lemma 4.4 (Ooto [Reference Ooto17, Lemma 7 and page 1058]).

Let $\alpha $ be a p-adic number with $|\alpha |_{p}\geq 1$ and let $p_{n}/q_{n}$ be the nth convergent of its Ruban p-adic continued fraction expansion. Then $p_{n}\leq |p_{n}|_{p}$ , $q_{n}\leq |q_{n}|_{p}$ and

$$ \begin{align*} p_{n}\cdot|p_{n}|_{p}\in\mathbb{Z} \quad q_{n}\cdot|q_{n}|_{p}\in\mathbb{Z}. \end{align*} $$

Theorem 4.5 (Lang [Reference Lang13, page 32]).

Let K be a p-adic algebraic number field and let $\alpha $ be any algebraic p-adic number. Then, for each $\varepsilon>0$ , the inequality

$$ \begin{align*} |\alpha-\beta|_{p}<\frac{1}{H(\beta)^{2+\varepsilon}} \end{align*} $$

has only finitely many solutions $\beta $ in K.

5. Proofs of Theorems 3.1 and 3.3

Proof of Theorem 3.1.

We prove Theorem 3.1 by adapting the method of the proof of Alnıaçık [Reference Alnıaçık2, Theorem] to the non-Archimedean p-adic case. Define the algebraic p-adic numbers

$$ \begin{align*} \alpha_{r_{n}}:=[b_{0}, b_{1},\ldots,b_{r_{n}},a_{r_{n}+1},a_{r_{n}+2},\ldots]_{p}\in\mathbb{Q}(\alpha) \quad (n=0,1,2,\ldots) \end{align*} $$

and

$$ \begin{align*} \beta_{r_{n}}:=[a_{r_{n}+1},a_{r_{n}+2},\ldots]_{p}\in\mathbb{Q}(\alpha) \quad (n=0,1,2,\ldots). \end{align*} $$

Then $\deg (\alpha _{r_{n}})=\deg (\beta _{r_{n}})=m\ (n=0,1,2,\ldots )$ . By (2.3),

$$ \begin{align*} \alpha=[a_{0}, a_{1},\ldots,a_{r_{n}},\beta_{r_{n}}]_{p}=\frac{p_{r_{n}}\beta_{r_{n}}+p_{r_{n}-1}}{q_{r_{n}}\beta_{r_{n}}+q_{r_{n}-1}} \quad (n=0,1,2,\ldots) \end{align*} $$

and thus

$$ \begin{align*} \alpha q_{r_{n}} \beta_{r_{n}}+\alpha q_{r_{n}-1}-p_{r_{n}}\beta_{r_{n}}-p_{r_{n}-1}=0 \quad (n=0,1,2,\ldots). \end{align*} $$

Therefore, $F(\beta _{r_{n}}, \alpha )=0$ , where, by Lemma 4.4,

$$ \begin{align*} F(x, x_{1})=|p_{r_{n}}|_{p}q_{r_{n}}x_{1}x+|p_{r_{n}}|_{p}q_{r_{n}-1}x_{1}-|p_{r_{n}}|_{p}p_{r_{n}}x-|p_{r_{n}}|_{p}p_{r_{n}-1} \end{align*} $$

is a polynomial in $x, x_{1}$ over $\mathbb {Z}$ . It follows from Theorem 4.2 and Lemma 4.4 that

(5.1) $$ \begin{align} H(\beta_{r_{n}})\leq c_{1}|q_{r_{n}}|_{p}^{2m}, \end{align} $$

where $c_{1}=3^{3m}|a_{0}|_{p}^{2m}H(\alpha )^{m}$ . Set

$$ \begin{align*} \frac{p_{n}'}{q_{n}'}:=[b_{0}, b_{1},\ldots,b_{n}]_{p} \quad (n=0,1,2,\ldots). \end{align*} $$

Then

$$ \begin{align*} \alpha_{r_{n}}=[b_{0}, b_{1},\ldots,b_{r_{n}},\beta_{r_{n}}]_{p}=\frac{p_{r_{n}}'\beta_{r_{n}}+p_{r_{n}-1}'}{q_{r_{n}}'\beta_{r_{n}}+q_{r_{n}-1}'} \quad (n=0,1,2,\ldots) \end{align*} $$

and

$$ \begin{align*} \alpha_{r_{n}} q_{r_{n}}' \beta_{r_{n}}+\alpha_{r_{n}} q_{r_{n}-1}'-p_{r_{n}}'\beta_{r_{n}}-p_{r_{n}-1}'=0 \quad (n=0,1,2,\ldots). \end{align*} $$

Thus, $F(\alpha _{r_{n}}, \beta _{r_{n}})=0$ , where, by Lemma 4.4,

$$ \begin{align*} F(x, x_{1})=|p_{r_{n}}'|_{p}q_{r_{n}}'x_{1}x+|p_{r_{n}}'|_{p}q_{r_{n}-1}'x-|p_{r_{n}}'|_{p}p_{r_{n}}'x_{1}-|p_{r_{n}}'|_{p}p_{r_{n}-1}' \end{align*} $$

is a polynomial in $x, x_{1}$ over $\mathbb {Z}$ . It follows from Theorem 4.2, Lemma 4.4 and (5.1) that

(5.2) $$ \begin{align} H(\alpha_{r_{n}})\leq 3^{3m}|p_{r_{n}}'|_{p}^{2m}c_{1}^{m}|q_{r_{n}}|_{p}^{2m^{2}}. \end{align} $$

From (3.4),

$$ \begin{align*} |b_{j}|_{p}\leq\kappa_{1}|a_{j}|_{p}^{\kappa_{2}} \quad (j=0,1,2,\ldots). \end{align*} $$

By Lemma 4.1,

(5.3) $$ \begin{align} |p_{r_{n}}'|_{p}\leq |a_{0}|_{p}^{\kappa_{2}}\kappa_{1}|q_{r_{n}}|_{p}^{\kappa_{2}+\log \kappa_{1}{/}\kern-1pt\log 2} \quad (n=0,1,2,\ldots). \end{align} $$

Using (5.2), (5.3) and $\lim _{n \rightarrow \infty } |q_{r_{n}}|_{p}=\infty $ , we obtain, for sufficiently large n,

(5.4) $$ \begin{align} H(\alpha_{r_{n}})\leq |q_{r_{n}}|_{p}^{c_{2}}, \end{align} $$

where $c_{2}=1+(m+\kappa _{2}+\log \kappa _{1}{/}\kern-1.2pt\log 2)2m$ .

We approximate $\xi $ by the algebraic p-adic numbers $\alpha _{r_{n}}$ . We infer from (2.6) and (3.4) that

(5.5) $$ \begin{align} |\xi-\alpha_{r_{n}}|_{p}\leq \max \bigg\{\bigg|\xi-\frac{p_{s_{n}}'}{q_{s_{n}}'}\bigg|_{p}, \bigg|\alpha_{r_{n}}-\frac{p_{s_{n}}'}{q_{s_{n}}'}\bigg|_{p}\bigg\}<\frac{1}{|q_{s_{n}}'|_{p}^{2}} \quad (n=0,1,2,\ldots). \end{align} $$

Put

$$ \begin{align*} \frac{d_{r_{n}}}{e_{r_{n}}}:=[a_{r_{n}+1},a_{r_{n}+2},\ldots,a_{s_{n}}]_{p}=[b_{r_{n}+1},b_{r_{n}+2},\ldots,b_{s_{n}}]_{p}. \end{align*} $$

We have

$$ \begin{align*} \frac{p_{s_{n}}}{q_{s_{n}}}=[a_{0},a_{1},\ldots,a_{r_{n}},a_{r_{n}+1},a_{r_{n}+2},\ldots,a_{s_{n}}]_{p} \end{align*} $$

and

$$ \begin{align*} \frac{p_{s_{n}}'}{q_{s_{n}}'}=[b_{0},b_{1},\ldots,b_{r_{n}},b_{r_{n}+1},b_{r_{n}+2},\ldots,b_{s_{n}}]_{p}. \end{align*} $$

Then

$$ \begin{align*} |q_{s_{n}}|_{p}=|a_{1}\cdots a_{r_{n}}|_{p}|a_{r_{n}+1}\cdots a_{s_{n}}|_{p}=|q_{r_{n}}|_{p}|a_{r_{n}+1}|_{p}|e_{r_{n}}|_{p} \end{align*} $$

and

$$ \begin{align*} |q_{s_{n}}'|_{p}=|b_{1}\cdots b_{r_{n}+1}|_{p}|b_{r_{n}+2}\cdots b_{s_{n}}|_{p}>|e_{r_{n}}|_{p}. \end{align*} $$

Therefore,

(5.6) $$ \begin{align} |q_{s_{n}}|_{p}<|a_{r_{n}+1}|_{p}|q_{r_{n}}|_{p}|q_{s_{n}}'|_{p} \quad (n=0,1,2,\ldots). \end{align} $$

It follows from Lemmas 4.3 and 4.4 that

(5.7) $$ \begin{align} \bigg|\alpha-\frac{p_{r_{n}}}{q_{r_{n}}}\bigg|_{p}\geq\frac{1}{c_{3}|q_{r_{n}}|_{p}^{2m}}, \end{align} $$

where $c_{3}=(m+1)2^{m}H(\alpha )|a_{0}|_{p}^{2m}$ . On the other hand, by (2.6),

(5.8) $$ \begin{align} \bigg|\alpha-\frac{p_{r_{n}}}{q_{r_{n}}}\bigg|_{p}=\frac{1}{|a_{r_{n}+1}|_{p}|q_{r_{n}}|_{p}^{2}} \quad (n=0,1,2,\ldots). \end{align} $$

Combining (5.6), (5.7) and (5.8), we get

(5.9) $$ \begin{align} |q_{s_{n}}|_{p}<c_{3}|q_{r_{n}}|_{p}^{2m-1}|q_{s_{n}}'|_{p}. \end{align} $$

By (3.2) and (5.9),

$$ \begin{align*} c_{3}|q_{r_{n}}|_{p}^{2m-1}\leq |q_{s_{n}}'|_{p} \end{align*} $$

for sufficiently large n. So, for sufficiently large n,

(5.10) $$ \begin{align} |q_{s_{n}}|_{p}<|q_{s_{n}}'|_{p}^{2}. \end{align} $$

We see from (3.2), (5.4), (5.5) and (5.10) that

$$ \begin{align*} 0<|\xi-\alpha_{r_{n}}|_{p}<\frac{1}{|q_{s_{n}}|_{p}}\leq\frac{1}{H(\alpha_{r_{n}})^{\phi_{n}}} \end{align*} $$

for sufficiently large n, where

$$ \begin{align*} \phi_{n}=\frac{\log |q_{s_{n}}|_{p}}{c_{2}\log |q_{r_{n}}|_{p}} \hspace{0.15in} \textrm{and} \hspace{0.15in} \lim_{n \rightarrow \infty} \phi_{n}=\infty. \end{align*} $$

As $\deg (\alpha _{r_{n}})=m\ (n=0,1,2,\ldots )$ , this shows that $\xi $ is a p-adic $U^{*}$ -number with

(5.11) $$ \begin{align} w_{m}^{*}(\xi)=\infty. \end{align} $$

We wish to show that $\xi $ is a p-adic $U_{m}^{*}$ -number. We must prove that $w_{t}^{*}(\xi )<\infty $ for $t=1,\ldots ,m-1$ . Let $\beta $ be any algebraic p-adic number with $1\leq \deg (\beta )\leq m-1$ and with sufficiently large height $H(\beta )$ . We deduce from Lemma 4.3 and (5.4) that

(5.12) $$ \begin{align} |\alpha_{r_{n}}-\beta|_{p}\geq \frac{1}{c_{4}|q_{r_{n}}|_{p}^{c_{5}}H(\beta)^{m}} \end{align} $$

for sufficiently large n, where $c_{4}=(m+1)^{m-1}m^{m}$ and $c_{5}=c_{2}(m-1)$ . By (3.3), there exists a real number $T>1$ such that

(5.13) $$ \begin{align} |q_{s_{n}}|_{p}^{T}\geq|q_{r_{n+1}}|_{p} \end{align} $$

for sufficiently large n. We have

(5.14) $$ \begin{align} |\xi-\beta|_{p}=|(\xi-\alpha_{r_{n}})+(\alpha_{r_{n}}-\beta)|_{p}. \end{align} $$

From (5.5), (5.10) and (5.13), for sufficiently large n,

(5.15) $$ \begin{align} |\xi-\alpha_{r_{n}}|_{p}<\frac{1}{|q_{s_{n}}'|_{p}^{2}}<\frac{1}{|q_{s_{n}}|_{p}}\leq\frac{1}{|q_{r_{n+1}}|_{p}^{1/T}}. \end{align} $$

Let i be the unique positive rational integer satisfying $|q_{r_{i}}|_{p}\leq H(\beta )<|q_{r_{i+1}}|_{p}$ . Put ${T_{1}:=T(m+c_{5}+1)}$ . If $|q_{r_{i}}|_{p}\leq H(\beta )<|q_{r_{i+1}}|_{p}^{1/T_{1}}$ , then it follows from (5.12), (5.14) and (5.15) with $n=i$ that

(5.16) $$ \begin{align} |\xi-\beta|_{p}\geq \frac{1}{c_{4}H(\beta)^{m+c_{5}}}. \end{align} $$

If $|q_{r_{i+1}}|_{p}^{1/T_{1}}\leq H(\beta )<|q_{r_{i+1}}|_{p}$ , then it follows from (3.2), (5.12), (5.14) and (5.15) with $n=i+1$ that

(5.17) $$ \begin{align} |\xi-\beta|_{p}\geq \frac{1}{c_{4}H(\beta)^{m+c_{5}T_{1}}}. \end{align} $$

We deduce from (5.16) and (5.17) that

$$ \begin{align*} |\xi-\beta|_{p}\geq \frac{1}{c_{4}H(\beta)^{m+c_{5}T_{1}}} \end{align*} $$

for all algebraic p-adic numbers $\beta $ with $\deg (\beta )\leq m-1$ and with sufficiently large height $H(\beta )$ . This gives

(5.18) $$ \begin{align} w_{t}^{*}(\xi)<\infty \quad (t=1,\ldots,m-1). \end{align} $$

We infer from (5.11) and (5.18) that $\xi $ is a p-adic $U_{m}^{*}$ -number. As the set of p-adic $U_{m}$ -numbers is equal to the set of p-adic $U_{m}^{*}$ -numbers, $\xi $ is a p-adic $U_{m}$ -number.

Example 5.1. This example illustrates Theorem 3.1. In Theorem 3.1, take the algebraic p-adic number $\alpha $ as the quadratic irrational

$$ \begin{align*} \alpha=[a_0,a_1,a_2,\ldots]_{p}=[p^{-2},p^{-2},p^{-2},\ldots]_{p} \end{align*} $$

and the sequences $(r_{n})_{n=0}^{\infty }$ and $(s_{n})_{n=0}^{\infty }$ as

$$ \begin{align*} r_{0}=0, r_{n}=2(n+1)! \ (n=1,2,3,\ldots) \quad\textrm{and} \quad s_{n}=(n+2)! \ (n=0,1,2,\ldots). \end{align*} $$

Define the rational numbers $b_{j}\ (j=0,1,2,\ldots )$ by

$$ \begin{align*} b_{j}= \begin{cases}p^{-2} \quad \mbox{if }r_{n}\leq j\leq s_{n} & (n=0,1,2,\ldots), \\ p^{-4} \quad \mbox{if }s_{n}<j<r_{n+1} & (n=0,1,2,\ldots). \end{cases} \end{align*} $$

Take $\kappa _{1}=1$ and $\kappa _{2}=2$ . Then all the conditions of Theorem 3.1 are satisfied and therefore the irrational p-adic number $\xi =[b_0,b_1,b_2,\ldots ]_{p}$ is a p-adic $U_{2}$ -number.

Remark 5.2. In Theorem 3.1, if we replace $\lim _{n \rightarrow \infty }(\log |q_{s_{n}}|_{p}{/}\kern-1.2pt\log |q_{r_{n}}|_{p})=\infty $ by

$$ \begin{align*} \liminf_{n \rightarrow \infty}\frac{\log |q_{s_{n}}|_{p}}{\log |q_{r_{n}}|_{p}}>T(1+m+c_{5}T_{1}) \quad \textrm{and} \quad \limsup_{n \rightarrow \infty}\frac{\log |q_{s_{n}}|_{p}}{\log |q_{r_{n}}|_{p}}=\infty, \end{align*} $$

then we see from the proof that Theorem 3.1 still holds true.

Proof of Theorem 3.3.

We replace (3.2) by (3.6) and keep all the lines of the proof of Theorem 3.1 up to (5.10). By (3.6), there exists a positive real number $\varepsilon $ such that

(5.19) $$ \begin{align} \frac{\log |q_{s_{n}}|_{p}}{\log |q_{r_{n}}|_{p}}>(2+\varepsilon)c_{2} \end{align} $$

for sufficiently large n. We deduce from (5.4), (5.5), (5.10) and (5.19) that

$$ \begin{align*} 0<|\xi-\alpha_{r_{n}}|_{p}<\frac{1}{H(\alpha_{r_{n}})^{2+\varepsilon}} \end{align*} $$

for sufficiently large n. It follows from the definition of $\alpha _{r_{n}}$ and (3.5) that the algebraic p-adic numbers $\alpha _{r_{n}}$ in $\mathbb {Q}(\alpha )$ are all distinct. Then, by Theorem 4.5, the irrational p-adic number $\xi $ is transcendental.

Finally, we pose the following question.