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Yann Bugeaud

Publications and source records attributed to Yann Bugeaud.

87 records · Page 5Linked to original sources
arXiv2007

Diophantine exponents for mildly restricted approximation

We are studying the Diophantine exponent μ_{n,l}$ defined for integers 1 \leq l < n and a vector α\in \mathbb{R}^n by letting μ_{n,l} = \sup{μ\geq 0: 0 < ||x \cdot α|| < H(x)^{-μ} for infinitely many x \in C_{n,l} \cap \mathbb{Z}^n}, where \cdot is the scalar product and || . || denotes the distance to the nearest integer and C_{n,l} is the generalised cone consisting of all vectors with the height attained among the first l coordinates. We show that the exponent takes all values in the interval [l+1, \infty), with the value n attained for almost all α. We calculate the Hausdorff dimension of the set of vectors αwith μ_{n,l} (α) = μfor μ\geq n. Finally, letting w_n denote the exponent obtained by removing the restrictions on x, we show that there are vectors αfor which the gaps in the increasing sequence μ_{n,1} (α) \leq ... \leq μ_{n,n-1} (α) \leq w_n (α) can be chosen to be arbitrary.

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arXiv2007

Sums of the error term function in the mean square for $ζ(s)$

Sums of the form $\sum_{n\le x}E^k(n) (k\in{\bf N}$ fixed) are investigated, where $$ E(T) = \int_0^T|ζ(1/2+it)|^2 dt - T\Bigl(\log {T\over2π} + 2γ-1\Bigr)$$ is the error term in the mean square formula for $|ζ(1/2+it)|$. The emphasis is on the case k=1, which is more difficult than the corresponding sum for the divisor problem. The analysis requires bounds for the irrationality measure of ${\rm e}^{2πm}$ and for the partial quotients in its continued fraction expansion.

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arXiv2006

Exponents of Diophantine approximation

The paper is mostly a survey on recent results in Diophantine approximation, with emphasis on properties of exponents measuring various notions of Diophantine <approximation.

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arXiv2005

Palindromic continued fractions

In the present work, we investigate real numbers whose sequence of partial quotients enjoys some combinatorial properties involving the notion of palindrome. We provide three new transendence criteria, that apply to a broad class of continued fraction expansions, including expansions with unbounded partial quotients. Their proofs heavily depend on the Schmidt Subspace Theorem.

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arXiv2005

On the complexity of algebraic number I. Expansions in integer bases

Let $b \ge 2$ be an integer. We prove that the $b$-adic expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion.

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arXiv2005

On the complexity of algebraic numbers II. Continued fractions

The continued fraction expansion of an irrational number $α$ is eventually periodic if and only if $α$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real numbers of degree at least three. Because of some numerical evidence and a belief that these numbers behave like most numbers in this respect, it is often conjectured that their partial quotients form an unbounded sequence. More modestly, we may expect that if the sequence of partial quotients of an irrational number $α$ is, in some sense, "simple", then $α$ is either quadratic or transcendental. The term "simple" can of course lead to many interpretations. It may denote real numbers whose continued fraction expansion has some regularity, or can be produced by a simple algorithm (by a simple Turing machine, for example), or arises from a simple dynamical system... The aim of this paper is to present in a unified way several new results on these different approaches of the notion of simplicity/complexity for the continued fraction expansion of algebraic real numbers of degree at least three.

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arXiv2005

On the Littlewood conjecture in simultaneous Diophantine approximation

For any given real number $α$ with bounded partial quotients, we construct explicitly continuum many real numbers $β$ with bounded partial quotients for which the pair $(α, β)$ satisfies a strong form of the Littlewood conjecture. Our proof is elementary and rests on the basic theory of continued fractions.

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arXiv2005

On the Littlewood conjecture in fields of power series

Let $\k$ be an arbitrary field. For any fixed badly approximable power series $Θ$ in $\k((X^{-1}))$, we give an explicit construction of continuum many badly approximable power series $Φ$ for which the pair $(Θ, Φ)$ satisfies the Littlewood conjecture. We further discuss the Littlewood conjecture for pairs of algebraic power series.

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arXiv2005

On the Maillet--Baker continued fractions

We use the Schmidt Subspace Theorem to establish the transcendence of a class of quasi-periodic continued fractions. This improves earlier works of Maillet and of A. Baker. We also improve an old result of Davenport and Roth on the rate of increase of the denominators of the convergents to any real algebraic number.

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arXiv2005

Continued fractions and transcendental numbers

It is widely believed that the continued fraction expansion of every irrational algebraic number $α$ either is eventually periodic (and we know that this is the case if and only if $α$ is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this question was first considered by Khintchine. A preliminary step towards its resolution consists in providing explicit examples of transcendental continued fractions. The main purpose of the present work is to present new families of transcendental continued fractions with bounded partial quotients. Our results are derived thanks to new combinatorial transcendence criteria recently obtained by Adamczewski and Bugeaud.

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arXiv2004

Exponents of Diophantine Approximation and Sturmian Continued Fractions

Let x be a real number and let n be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents w_n(x) and w_n^*(x) defined by Mahler and Koksma. We calculate their six values when n=2 and x is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction x by quadratic surds.

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arXiv2004

Exponents of inhomogeneous Diophantine Approximation

In Diophantine approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that the exponent of approximation to a generic point in R^n by a system of n linear forms is equal to the inverse of the uniform homogeneous exponent associated to the system of dual forms.

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arXiv2004

Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers

This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8, 144 and the only perfect powers in the Lucas sequence are 1, 4.

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