Transcendence Measures for some $U_m$-numbers related to Liouville's constant

TL;DR

This paper proves that multiplying or adding an algebraic number by the Liouville constant results in a $U$-number with a specific type, and provides bounds on certain transcendence measures related to these numbers.

Contribution

It establishes that algebraic numbers combined with the Liouville constant produce $U$-numbers with predictable types and bounds, advancing understanding of their transcendence properties.

Findings

Sum and product with Liouville constant produce $U$-numbers.

Type of resulting $U$-numbers equals the degree of the algebraic number.

Provides explicit bounds on transcendence measures for these numbers.

Abstract

In this note, we shall prove that the sum and the product of an algebraic number $\alpha$ by the \textit{Liouville constant} $L=\sum_{j=1}^{\infty}10^{-j!}$ is a $U$-number with type equals to the degree of $\alpha$ (with respect to $\mathbb{Q}$). Moreover, we shall have that $\max\{w^{\ast}_n(\alpha L),w^{\ast}_n(\alpha + L)\}\leq 2m^2n+m-1$, for $n=1,...,m-1$.

Equations

$$\max \{w_n^{\ast }(\alpha L),w_n^{\ast }(\alpha +L)\}\le 2m^{2}n+m-1,forn=1,...,m-1.$$

$$|L-{\alpha }_k|<\frac{10}{9}H({\alpha }_k{)}^{-k-1}.$$

$$|\alpha L-\alpha {\alpha }_k|\le cH({\alpha }_k{)}^{-k-1},$$

$$|\alpha L-\alpha {\alpha }_k|\le cH(\alpha {)}^{k+1}H({\gamma }_k{)}^{-k-1}.$$

$$H({\gamma }_{k+1})\le H(\alpha )H({\alpha }_{k+1}{)}^m=H(\alpha )H({\alpha }_k{)}^{(k+1)m}\le H(\alpha )H({\gamma }_k{)}^{(k+1)m}$$

$$H({\gamma }_k)<H(\gamma {)}^{2m^2}<H({\gamma }_{k+1})\le H(\alpha )H({\gamma }_k{)}^{(k+1)m}.$$

$$|{\gamma }_k-\gamma |\ge f(m,n)H(\gamma {)}^{-m}H({\gamma }_k{)}^{-n},$$

$$|{\gamma }_k-\gamma |\ge f(m,n)H(\alpha {)}^{-1/2m}H({\gamma }_k{)}^{-(k+1)/2-n}.$$

$$H({\gamma }_k{)}^{(k+1)/2-n}\ge 2cf(m,n{)}^{-1}H(\alpha {)}^{k+1/2m}.$$

$$|\alpha L-\gamma |$$

$$\max \{w_n^{\ast }(\alpha \beta ),w_n^{\ast }(\alpha +\beta )\}\le 2m^{2}n+m-1,forn=1,...,m-1.$$

Full text

$\max\{w^{\ast}_{n}(\alpha L),w^{\ast}_{n}(\alpha+L)\}\leq 2m^{2}n+m-1$, for $n=1,...,m-1$.

1. Introduction

A real number $\xi$ is called a Liouville number, if for any positive real number $w$ there exist infinitely many rational numbers $p/q$, with $q\geq 1$, such that

$0<\left|\xi-\dfrac{p}{q}\right|<\dfrac{1}{q^{w}}$.

Transcendental number theory began in 1844 when Liouville [6] showed that all Liouville numbers are transcendental establishing thus the first examples of such numbers. For instance, the number

$L=\displaystyle\sum_{j=1}^{\infty}10^{-j!}=0.11000100000000000000001000...$,

which is known as Liouville’s constant, is a Liouville number and therefore transcendental. In 1962, Erd$\ddot{\mbox{o}}$s [3] proved that every nonzero real number can be written as the sum and the product of two Liouville numbers.

In 1932, Mahler [7] splited the set of the transcendental numbers in three disjoint sets named $S$-, $T$- and $U$-numbers. Particularly, the $U$-numbers generalizes the concept of Liouville numbers. We denote by $w^{*}_{n}(\xi)$ as the supremum of the real numbers $w^{*}$ for which there exist infinitely many real algebraic numbers $\alpha$ of degree $n$ satisfying

$0<|\xi-\alpha|<H(\alpha)^{-w^{*}-1}$,

where $H(\alpha)$ (so-called the height of $\alpha$) is the maximum of absolute value of coefficients of the minimal polynomial of $\alpha$ (over $\mathbb{Z}$). The number $\xi$ is said to be a $U^{*}_{m}$-number (according to LeVeque [5]) if $w^{*}_{m}(\xi)=\infty$ and $w^{*}_{n}(\xi)<\infty$ for $1\leq n<m$ ($m$ is called the type of the $U$-number). We point out that we actually have defined a Koksma $U^{*}_{m}$-number instead of a Mahler $U_{m}$-number. However, it is well-known that they are the same [2, cf. Theorem 3.6] and [1]. We observe that the set of $U_{1}$-numbers is precisely the set of Liouville numbers.

The existence of $U_{m}$-numbers, for all $m\geq 1$, was proved by LeVeque [5]. In 1993, Pollington [8] showed that for any positive integer $m$, every real number can be expressed as a sum of two $U_{m}$-numbers.

Since two algebraically dependent numbers must belong to the same Mahler’s class [2, Theorem 3.2], then $\alpha L$ and $\alpha+L$ are $U$-numbers, for any nonzero algebraic number $\alpha$. But what are their types?

In this note, we use G$\ddot{\mbox{u}}$tting’s method [4] for proving that the sum and the product of every $m$-degree algebraic number $\alpha$ by $L$ is a $U_{m}$-number. Moreover, we obtain an upper bound for $w_{n}^{\ast}$.

Theorem 1**.**

Let $\alpha$ be an algebraic number of degree $m$ and let $L$ be the Liouville’s constant. Then $\alpha L$ and $\alpha+L$ are $U_{m}$-numbers, with

[TABLE]

2. Auxiliary Results

Before the proof of the main result, we need two technical results. The first one follows as an easy consequence of the triangular inequality and binomial identities.

Lemma 1**.**

Given $P(x)\in\mathbb{Z}[x]$ with degree $m$ and $a/b\in\mathbb{Q}\backslash\{0\}$. If $Q_{1}(x)=a^{m}P(\frac{b}{a}x)$ and $Q_{2}(x)=b^{m}P(x-\frac{a}{b})$, then

• (i)

$H(Q_{1})\leq\max\{|a|,|b|\}^{m}H(P)$;

• (ii)

$H(Q_{2})\leq 2^{m+1}\max\{|a|,|b|\}^{m}H(P)$.

Proof.

(i) If $P(x)=\sum_{j=0}^{m}a_{j}x^{j}$, then $Q_{1}(x)=\sum_{j=0}^{m}a_{j}b^{j}a^{m-j}x^{j}$. Supposing, without loss of generality, that $|a|\geq|b|$, we have $|a|^{m}|a_{j}|\geq|a|^{m-j}|a_{j}||b|^{j}$ for $0\leq j\leq m$. Hence, we are done. For (ii), write $Q_{2}(x)=\sum_{i=0}^{m}c_{i}x^{i}$, where

$c_{i}=b^{m}\displaystyle\sum_{j=i}^{m}a_{j}{j\choose j-i}(-1)^{j-i}\left(\frac{a}{b}\right)^{j-i}$

Therefore

$|c_{i}|\leq H(P)\displaystyle\sum_{k=0}^{m-i}\displaystyle{k+i\choose k}|a|^{k}|b|^{m-k}\leq\max\{|a|,|b|\}^{m}H(P)\displaystyle\sum_{k=0}^{m-i}\displaystyle{k+i\choose k}$.

Since $\sum_{k=0}^{m-i}{k+i\choose k}={m+1\choose m-i}\leq 2^{m+1}$, we finally have

$|c_{i}|\leq 2^{m+1}\max\{|a|,|b|\}^{m}H(P)$,

which completes our proof. ∎

In addition to Lemma 1, we use the fact that algebraic numbers are not well aproximable by algebraic numbers.

Lemma 2** (Cf. Corollary A.2 of [2]).**

Let $\alpha$ and $\beta$ be two distinct nonzero algebraic numbers of degree $n$ and $m$, respectively. Then we have

$|\alpha-\beta|\geq(n+1)^{-m/2}(m+1)^{-n/2}\max\{2^{-n}(n+1)^{-(m-1)/2},2^{-m}(m+1)^{-(n-1)/2}\}$

$\times H(\alpha)^{-m}H(\beta)^{-n}$.

Proof.

A sketch of the proof can be found in the Appendix A of [2]. ∎

3. Proof of the Theorem

For $k\geq 1$, set

$p_{k}=10^{k!}\displaystyle\sum_{j=1}^{k}10^{-j!}$, $q_{k}=10^{k!}$ and $\alpha_{k}=\dfrac{p_{k}}{q_{k}}$.

We observe that $H(\alpha_{k-1})<H(\alpha_{k})=10^{k!}=H(\alpha_{k-1})^{k}$ and

[TABLE]

Thus, setting $\gamma_{k}=\alpha\alpha_{k}$, we obtain of (3.1)

[TABLE]

where $c=10|\alpha|/9$. It follows by the Lemma 1 (i) that $H(\alpha_{k})^{m}\geq H(\alpha)^{-1}H(\gamma_{k})$ and thus we conclude that

[TABLE]

Consequently, $\alpha\beta$ is a $U$-number with type at most $m$ (since $\gamma_{k}$ has degree $m$).

Again, we use Lemma 1 (i) for obtaining

[TABLE]

Now, let $\gamma$ be an $n$-degree real algebraic number, with $n<m$ and $H(\gamma)\geq H(\gamma_{1})$. Thus, one may ensure the existence of a sufficient large $k$ such that

[TABLE]

So, by Lemma 2, it follows that

[TABLE]

where $f(m,n)$ is a positive number which does not depend on $k$ and $\gamma$. Therefore by (3.5)

[TABLE]

By taking $H(\gamma)$ large enough, the index $k$ satisfies

[TABLE]

Thus, it follows from (3.3), (3.7) and (3.8) that $|\gamma_{k}-\gamma|\geq 2|\alpha L-\gamma_{k}|$. Therefore, except for finitely many algebraic numbers $\gamma$ of degree $n$ strictly less than $m$, we have

[TABLE]

where we use the left-hand side of (3.5). It follows that $w_{n}^{*}(\alpha L)\leq 2m^{2}n+m-1$ which finishes our proof.

The case $\alpha+L$ follows the same outline, where we use Lemma 1 (ii) rather than (i). ∎

4. The general case and further comments

Let $\beta$ be a Liouville number. Since that a $U$-number keeps its type when multiplied by any nonzero rational number, we can consider $0<\beta<1$. Set

$S_{\beta}=\{(\frac{p_{k}}{q_{k}})_{k\geq 1}\in\mathbb{Q}^{\infty}\ :\ |\beta-\frac{p_{k}}{q_{k}}|<\frac{1}{q_{k}^{k+1}},\ k=1,2,...\}$.

By the assumption on $\beta$, we may suppose $1\leq p_{k}\leq q_{k}$ and then $H(p_{k}/q_{k})=q_{k}$, for all $k$. Note that $S_{\beta}$ is an infinite set.

As is customary, the symbols $\ll$, $\gg$ mean that there is an implied constant in the inequalities $\leq$, $\geq$, respectively. In our process for proving the Theorem 1, the key step happens when holds an inequality like in (3.5). Thus it follows that

Theorem 2**.**

Let $\alpha$ be an $m$-degree algebraic number and let $\beta$ be a Liouville number. If there exists a sequence $(p_{k}/q_{k})_{k\geq 1}\in S_{\beta}$ such that $q_{k}\ll q_{k+1}\ll q_{k}^{k+1}$ for all $k\gg 1$, then the numbers $\alpha\beta$ and $\alpha+\beta$ are $U_{m}$-numbers and

[TABLE]

Example 1**.**

For any integer number $m\geq 2$ and any $a_{j}\in\{1,...,9\}$, the number $\sum_{j=1}^{\infty}a_{j}m^{-j!}$ is a Liouville number satisfying the hypothesis of the previous theorem.

Corollary 1**.**

For any $m\geq 1$, there exists an uncountable collection of Liouville numbers that are expressible as sum of two algebraically dependent $U_{m}$-numbers.

Proof.

Set $\beta=\displaystyle\sum_{j=1}^{\infty}a_{j}10^{-j!}$, where $a_{j}\in\{1,2\}$. The result follows immediately of Theorem 2 and of writing $\beta=(\frac{\beta+\sqrt[m]{2}}{2})+(\frac{\beta-\sqrt[m]{2}}{2})$. ∎

There exist several lower estimates for the distance between two distinct algebraic numbers, e.g., Liouville’s inequality and Lemma 2. A too-good-to-be-true Conjecture due to Schmidt [9] states that

Conjecture 1**.**

For any number field $\mathbb{K}$ and any positive real number $\epsilon$, we have

$|\alpha-\beta|>c(\mathbb{K},\epsilon)(\max\{H(\alpha),H(\beta)\})^{-2-\epsilon}$,

for any distinct $\alpha,\beta\in\mathbb{K}$, where $c(\mathbb{K},\epsilon)$ is some constant depending only on $\mathbb{K}$ and on $\epsilon$.

We conclude by pointing that if the Schmidt’s conjecture is true, then the sum and the product of any $m$-degree algebraic number $\alpha$ by any Liouville number $\beta$ is a $U_{m}$-number and the inequality (4.1) can be considerable improved for

$\max\{w^{\ast}_{n}(\alpha\beta),w^{\ast}_{n}(\alpha+\beta)\}\leq 1$.

Acknowledgement

The second author thanks to the Department of Mathematics of Universidade Federal do Ceará for its hospitality. We also thank Yann Bugeaud for nice discussions on the subject.