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Yann Bugeaud

Publications and source records attributed to Yann Bugeaud.

At least 37 records · Page 2Linked to original sources
arXiv2021

On the rational approximation to $p$-adic Thue--Morse numbers

Let $p$ be a prime number and $ξ$ an irrational $p$-adic number. Its multiplicative irrationality exponent ${μ^{\times}} (ξ)$ is the supremum of the real numbers ${μ^{\times}}$ for which the inequality $$ |b ξ- a|_{p} \leq | a b |^{- {μ^{\times}} / 2} $$ has infinitely many solutions in nonzero integers $a, b$. We show that ${μ^{\times}} (ξ)$ can be expressed in terms of a new exponent of approximation attached to a sequence of rational numbers defined in terms of $ξ$. We establish that ${μ^{\times}} ({{ξ_{{\bf t}, p}}}) = 3$, where ${{ξ_{{\bf t}, p}}}$ is the $p$-adic number $1 - p - p^2 + p^3 - p^4 + \ldots$, whose sequence of digits is given by the Thue--Morse sequence over $\{-1, 1\}$.

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arXiv2021

On the Lévy constants of Sturmian continued fractions

The Lévy constant of an irrational real number is defined by the exponential growth rate of the sequence of denominators of the principal convergents in its continued fraction expansion. Any quadratic irrational has an ultimately periodic continued fraction expansion and it is well-known that this implies the existence of a Lévy constant. Let $a, b$ be distinct positive integers. If the sequence of partial quotients of an irrational real number is a Sturmian sequence over $\{a, b\}$, then it has a Lévy constant, which depends only on $a$, $b$, and the slope of the Sturmian sequence, but not on its intercept. We show that the set of Lévy constants of irrational real numbers whose sequence of partial quotients is periodic or Sturmian is equal to the whole interval $[\log ((1+\sqrt 5)/2 ), + \infty)$.

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arXiv2021

On the rational approximation to Thue--Morse rational numbers

Let $b \ge 2$ and $\ell \ge 1$ be integers. We establish that there is an absolute real number $K$ such that all the partial quotients of the rational number $$ \prod_{h = 0}^\ell \, (1 - b^{-2^h}), $$ of denominator $b^{2^{\ell+1} - 1}$, do not exceed $\exp(K (\log b)^2 \sqrt{\ell} 2^{\ell/2})$.

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arXiv2021

Switched server systems whose parameters are normal numbers in base 4

Switched server systems are mathematical models of manufacturing, traffic and queueing systems. Recently, it was proved in (Eur. J. Appl. Math. 31(4) (2020), pp. 682-708) that there exist switched server systems with $3$ buffers (tanks), a server, filling rates $ρ_1=ρ_2=ρ_3=\frac13$ and parameters $d_1, d_2, d_3>0$ whose global attractor is a fractal set. In this article, we prove that if $x_1$ in $(0,\frac13)$, $x_2$ in $(\frac13,\frac23)$ and $x_3$ in $(\frac23,1)$ are rational numbers or normal numbers in base $4$ (or more generally, rich numbers to base $4$) and $(d_1,d_2,d_3)$ is the vector with positive entries satisfying $$d_1=\frac{1}{3x_1}-1,\quad d_2=\frac{2-3x_2}{3x_2-1}, \quad d_3=\frac{3-3x_3}{3x_3-2},$$ then the corresponding switched server has no fractal attractor. More precisely, the Poincaré map of the system has a finite global attractor. The approach we use is to study the topological dynamics of a family of piecewise $λ$-affine contractions that includes the Poincaré map of the switched server system as a particular case.

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arXiv2021

Classical and uniform exponents of multiplicative $p$-adic approximation

Let $p$ be a prime number and $ξ$ an irrational $p$-adic number. Its irrationality exponent $μ(ξ)$ is the supremum of the real numbers $μ$ for which the system of inequalities $$ 0 < \max\{|x|, |y|\} \le X, \quad |y ξ- x|_{p} \leq X^{-\hmu} $$ has a solution in integers $x, y$ for arbitrarily large real number $X$. Its multiplicative irrationality exponent $\tmu (ξ)$ (resp., uniform multiplicative irrationality exponent $\htmu (ξ)$) is the supremum of the real numbers $\hmu$ for which the system of inequalities $$ 0 < |x y|^{1/2} \le X, \quad |y ξ- x|_{p} \leq X^{-\hmu} $$ has a solution in integers $x, y$ for arbitrarily large (resp., for every sufficiently large) real number $X$. It is not difficult to show that $μ(ξ) \le \tmu(ξ) \le 2 μ(ξ)$ and $\htmu (ξ) \le 4$. We establish that the ratio between the multiplicative irrationality exponent $\tmu$ and the irrationality exponent $μ$ can take any given value in $[1, 2]$. Furthermore, we prove that $\htmu (ξ) \le (5 + \sqrt{5})/2$ for every $p$-adic number $ξ$.

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arXiv2021

Combinatorial structure of Sturmian words and continued fraction expansions of Sturmian numbers

Let $θ= [0; a_1, a_2, \dots]$ be the continued fraction expansion of an irrational real number $θ\in (0, 1)$. It is well-known that the characteristic Sturmian word of slope $θ$ is the limit of a sequence of finite words $(M_k)_{k \ge 0}$, with $M_k$ of length $q_k$ (the denominator of the $k$-th convergent to $θ$) being a suitable concatenation of $a_k$ copies of $M_{k-1}$ and one copy of $M_{k-2}$. Our first result extends this to any Sturmian word. Let $b \ge 2$ be an integer. Our second result gives the continued fraction expansion of any real number $ξ$ whose $b$-ary expansion is a Sturmian word ${\bf s}$ over the alphabet $\{0, b-1\}$. This extends a classical result of Böhmer who considered only the case where ${\bf s}$ is characteristic. As a consequence, we obtain a formula for the irrationality exponent of $ξ$ in terms of the slope and the intercept of ${\bf s}$.

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arXiv2021

Apollonius circles and irreducibility criteria for polynomials

We prove the irreducibility of integer polynomials $f(X)$ whose roots lie inside an Apollonius circle associated to two points on the real axis with integer abscisae $a$ and $b$, with ratio of the distances to these points depending on the canonical decomposition of $f(a)$ and $f(b)$. In particular, we obtain irreducibility criteria for the case where $f(a)$ and $f(b)$ have few prime factors, and $f$ is either an Eneström-Kakeya polynomial, or has a large leading coefficient. Analogous results are also provided for multivariate polynomials over arbitrary fields, in a non-Archimedean setting.

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arXiv2021

On simultaneous rational approximation to a $p$-adic number and its integral powers, II

Let $p$ be a prime number. For a positive integer $n$ and a real number $ξ$, let $λ_n (ξ)$ denote the supremum of the real numbers $λ$ for which there are infinitely many integer tuples $(x_0, x_1, \ldots , x_n)$ such that $| x_0 ξ- x_1|_p, \ldots , | x_0 ξ^n - x_n|_p$ are all less than $X^{-λ- 1}$, where $X$ is the maximum of $|x_0|, |x_1|, \ldots , |x_n|$. We establish new results on the Hausdorff dimension of the set of real numbers $ξ$ for which $λ_n (ξ)$ is equal to (or greater than or equal to) a given value.

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arXiv2020

Mahler's and Koksma's classifications in fields of power series

Let $q$ a prime power and ${\mathbb F}_q$ the finite field of $q$ elements. We study the analogues of Mahler's and Koksma's classifications of complex numbers for power series in ${\mathbb F}_q((T^{-1}))$. Among other results, we establish that both classifications coincide, thereby answering a question of Ooto.

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arXiv2019

Absolute root separation

The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height (the maximum absolute value of the coefficients) of the polynomial. We improve the known bounds for this problem and related ones. Then we report on extensive experiments in low degrees, suggesting that the current bounds are still very pessimistic.

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arXiv2019

On the Zeckendorf representation of smooth numbers

Among other results, we establish, in a quantitative form, that any sufficiently large integer cannot simultaneously be divisible only by very small primes and have very few digits in its Zeckendorf representation.

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arXiv2019

Inhomogeneous Diophantine approximation over fields of formal power series

We prove a sharp analogue of Minkowski's inhomogeneous approximation theorem over fields of power series $\mathbb{F}_q((T^{-1}))$. Furthermore, we study the approximation to a given point $\underline{y}$ in $\mathbb{F}_q((T^{-1}))^2$ by the $SL_2(\mathbb{F}_q[T])$-orbit of a given point $\underline{x}$ in $\mathbb{F}_q((T^{-1}))^2$.

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arXiv2019

Effective simultaneous rational approximation to pairs of real quadratic numbers

Let $ξ, ζ$ be quadratic real numbers in distinct quadratic fields. We establish the existence of effectively computable, positive real numbers $τ$ and $c$, such that, for every integer $q$ with $q > c$ we have $$ \max\{\|q ξ\|, \|q ζ\| \} > q^{-1 + τ}, $$ where $\| \cdot \|$ denotes the distance to the nearest integer.

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arXiv2019

On homogeneous and inhomogeneous Diophantine approximation over the fields of formal power series

We prove over fields of power series the analogues of several Diophantine approximation results obtained over the field of real numbers. In particular we establish the power series analogue of Kronecker's theorem for matrices, together with a quantitative form of it, which can also be seen as a transference inequality between uniform approximation and inhomogeneous approximation. Special attention is devoted to the one dimensional case. Namely, we give a necessary and sufficient condition on an irrational power series $α$ which ensures that, for some positive $\eps$, the set $$ \liminf_{Q \in \mathbb{F}_q[z], \,\, °Q \to \infty} \| Q \| \cdot |\langle Q α- θ\rangle| \geq \eps $$ has full Hausdorff dimension.

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arXiv2019

On simultaneous rational approximation to a real number and its integral powers, II

For a positive integer $n$ and a real number $ξ$, let $λ_n (ξ)$ denote the supremum of the real numbers $λ$ for which there are arbitrarily large positive integers $q$ such that $|| q ξ||, || q ξ^2 ||, \ldots , ||q ξ^n||$ are all less than $q^{-λ}$. Here, $|| \cdot ||$ denotes the distance to the nearest integer. We establish new results on the Hausdorff dimension of the set of real numbers $ξ$ such that $λ_n (ξ)$ is equal (or greater than or equal) to a given value.

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arXiv2018

Schmidt's game, fractals, and numbers normal to no base

Given $b > 1$ and $y \in \mathbb{R}/\mathbb{Z}$, we consider the set of $x\in \mathbb{R}$ such that $y$ is not a limit point of the sequence $\{b^n x \bmod 1: n\in\mathbb{N}\}$. Such sets are known to have full Hausdorff dimension, and in many cases have been shown to have a stronger property of being winning in the sense of Schmidt. In this paper, by utilizing Schmidt games, we prove that these sets and their bi-Lipschitz images must intersect with `sufficiently regular' fractals $K\subset \mathbb{R}$ (that is, supporting measures $μ$ satisfying certain decay conditions). Furthermore, the intersection has full dimension in $K$ if $μ$ satisfies a power law (this holds for example if $K$ is the middle third Cantor set). Thus it follows that the set of numbers in the middle third Cantor set which are normal to no base has dimension $\log2/\log3$.

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