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Yann Bugeaud

Publications and source records attributed to Yann Bugeaud.

At least 73 records · Page 4Linked to original sources
arXiv2015

On the complexity of a putative counterexample to the $p$-adic Littlewood conjecture

Let $|| \cdot ||$ denote the distance to the nearest integer and, for a prime number $p$, let $| \cdot |_p$ denote the $p$-adic absolute value. In 2004, de Mathan and Teulié asked whether $\inf_{q \ge 1} \, q \cdot || q α|| \cdot | q |_p = 0$ holds for every badly approximable real number $α$ and every prime number $p$. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number $α$ grows too rapidly or too slowly, then their conjecture is true for the pair $(α, p)$ with $p$ an arbitrary prime.

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arXiv2014

The Irrationality Exponents of Computable Numbers

We prove that a real number a greater than or equal to 2 is the irrationality exponent of some computable real number if and only if a is the upper limit of a computable sequence of rational numbers. Thus, there are computable real numbers whose irrationality exponent is not computable.

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arXiv2014

A combinatorial proof of the non-vanishing of Hankel determinants of the Thue--Morse sequence

In 1998, Allouche, Peyrière, Wen and Wen established that the Hankel determinants associated with the Thue--Morse sequence on $\{-1, 1\}$ are always nonzero. Their proof depends on a set of sixteen recurrence relations. We present an alternative, purely combinatorial proof of the same result. We also re-prove a recent result of Coons on the non-vanishing of the Hankel determinants associated to two other classical integer sequences.

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arXiv2014

Uniform Diophantine approximation related to $b$-ary and $β$-expansions

Let $b\geq 2$ be an integer and $\hv$ a real number. Among other results, we compute the Hausdorff dimension of the set of real numbers $ξ$ with the property that, for every sufficiently large integer $N$, there exists an integer $n$ such that $1 \le n \le N$ and the distance between $b^n ξ$ and its nearest integer is at most equal to $b^{-\hv N}$. We further solve the same question when replacing $b^nξ$ by $T^n_βξ$, where $T_β$ denotes the classical $β$-transformation.

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arXiv2013

On Simply Normal Numbers to Different Bases

Let s be an integer greater than or equal to 2. A real number is simply normal to base s if in its base-s expansion every digit 0, 1, ..., s-1 occurs with the same frequency 1/s. Let X be the set of positive integers that are not perfect powers, hence X is the set {2,3, 5,6,7,10,11,...} . Let M be a function from X to sets of positive integers such that, for each s in X, if m is in M(s) then each divisor of m is in M(s) and if M(s) is infinite then it is equal to the set of all positive integers. These conditions on M are necessary for there to be a real number which is simply normal to exactly the bases s^m such that s is in X and m is in M(s). We show these conditions are also sufficient and further establish that the set of real numbers that satisfy them has full Hausdorff dimension. This extends a result of W. M. Schmidt (1961/1962) on normal numbers to different bases.

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arXiv2013

Metric Diophantine approximation on the middle-third Cantor set

Let $μ\geq 2$ be a real number and let $\Mcal(μ)$ denote the set of real numbers approximable at order at least $μ$ by rational numbers. More than eighty years ago, Jarní k and, independently, Besicovitch established that the Hausdorff dimension of $\Mcal(μ)$ is equal to $2/μ$. We investigate the size of the intersection of $\Mcal(μ)$ with Ahlfors regular compact subsets of the interval $[0, 1]$. In particular, we propose a conjecture for the exact value of the dimension of $\Mcal(μ)$ intersected with the middle-third Cantor set and give several results supporting this conjecture. We especially show that the conjecture holds for a natural probabilistic model that is intended to mimic the distribution of the rationals. The core of our study relies heavily on dimension estimates concerning the set of points lying in an Ahlfors regular set and approximated at a given rate by a system of random points.

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arXiv2012

Automatic continued fractions are transcendental or quadratic

We establish new combinatorial transcendence criteria for continued fraction expansions. Let $α= [0; a_1, a_2,...]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients $(a_{\ell})_{\ell \ge 1}$ of $α$ cannot be generated by a finite automaton, and that the complexity function of $(a_{\ell})_{\ell \ge 1}$ cannot increase too slowly.

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arXiv2011

Root separation for irreducible integer polynomials

We establish new results on root separation of integer, irreducible polynomials of degree at least four. These improve earlier bounds of Bugeaud and Mignotte (for even degree) and of Beresnevich, Bernik, and Goetze (for odd degree).

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arXiv2010

On fractional parts of powers of real numbers close to 1

We prove that there exist arbitrarily small positive real numbers $ε$ such that every integral power $(1 + \vepsilon)^n$ is at a distance greater than $2^{-17} ε|\log \vepsilon|^{-1}$ to the set of rational integers. This is sharp up to the factor $2^{-17} |\log ε|^{-1}$. We also establish that the set of real numbers $α> 1$ such that the sequence of fractional parts $(\{α^n\})_{n \ge 1}$ is not dense modulo 1 has full Hausdorff dimension.

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arXiv2010

Transcendence with Rosen continued fractions

We give the first transcendence results for the Rosen continued fractions. Introduced over half a century ago, these fractions expand real numbers in terms of certain algebraic numbers.

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arXiv2010

Metric considerations concerning the mixed Littlewood Conjecture

The main goal of this note is to develop a metrical theory of Diophantine approximation within the framework of the de Mathan-Teulie Conjecture, also known as the `Mixed Littlewood Conjecture'. Let p be a prime. A consequence of our main result is that, for almost every real number α, \liminf_{n\rar\infty}n(\log n)^2|n|_p\|nα\|=0.

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arXiv2009

Badly approximable numbers and Littlewood-type problems

We establish that the set of pairs $(α, β)$ of real numbers such that $$ \liminf_{q \to + \infty} q \cdot (\log q)^2 \cdot \Vert q α\Vert \cdot \Vert q β\Vert > 0, $$ where $\Vert \cdot \Vert$ denotes the distance to the nearest integer, has full Hausdorff dimension in $\R^2$. Our proof rests on a method introduced by Peres and Schlag, that we further apply to various Littlewood-type problems

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arXiv2009

Morphic and Automatic Words: Maximal Blocks and Diophantine Approximation

Let $\mb w$ be a morphic word over a finite alphabet $Σ$, and let $Δ$ be a nonempty subset of $Σ$. We study the behavior of maximal blocks consisting only of letters from $Δ$ in $\mb w$, and prove the following: let $(i_k,j_k)$ denote the starting and ending positions, respectively, of the $k$'th maximal $Δ$-block in $\mb w$. Then $\limsup_{k\to\infty} (j_k/i_k)$ is algebraic if $\mb w$ is morphic, and rational if $\mb w$ is automatic. As a result, we show that the same conclusion holds if $(i_k,j_k)$ are the starting and ending positions of the $k$'th maximal zero block, and, more generally, of the $k$'th maximal $x$-block, where $x$ is an arbitrary word. This enables us to draw conclusions about the irrationality exponent of automatic and morphic numbers. In particular, we show that the irrationality exponent of automatic (resp., morphic) numbers belonging to a certain class that we define is rational (resp., algebraic).

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arXiv2009

On a mixed problem in Diophantine approximation

Let $d$ be a positive integer. Let $p$ be a prime number. Let $α$ be a real algebraic number of degree $d+1$. We establish that there exist a positive constant $c$ and infinitely many algebraic numbers $ξ$ of degree $d$ such that $|α- ξ| \cdot \min\{|\Norm(ξ)|_p,1\} < c H(ξ)^{-d-1} (\log 3 H(ξ))^{-1/d}$. Here, $H(ξ)$ and $\Norm(ξ)$ denote the na{\"ı}ve height of $ξ$ and its norm, respectively. This extends an earlier result of de Mathan and Teulié that deals with the case $d=1$.

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arXiv2008

On shrinking targets for Z^m actions on tori

Let A be an n by m matrix with real entries. Consider the set Bad_A of x \in [0,1)^n for which there exists a constant c(x)>0 such that for any q \in Z^m the distance between x and the point {Aq} is at least c(x) |q|^{-m/n}. It is shown that the intersection of Bad_A with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new.

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arXiv2007

Approximation of complex algebraic numbers by algebraic numbers of bounded degree

We investigate how well complex algebraic numbers can be approximated by algebraic numbers of degree at most n. We also investigate how well complex algebraic numbers can be approximated by algebraic integers of degree at most n+1. It follows from our investigations that for every positive integer n there are complex algebraic numbers of degree larger than n that are better approximable by algebraic numbers of degree at most n than almost all complex numbers. As it turns out, these numbers are more badly approximable by algebraic integers of degree at most n+1 than almost all complex numbers.

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arXiv2007

On two notions of complexity of algebraic numbers

we derive new, improved lower bounds for the block complexity of an irrational algebraic number and for the number of digit changes in the b-ary expansion of an irrational algebraic number. To this end, we apply a quantitative version of the Subspace Theorem due to Evertse and Schlickewei (2002).

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