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In the previous section we used (but did not prove) Stickelberger’s theorem, which provides a nontrivial annihilator for the class group. In this chapter we prove this theorem, in a stronger form: we define an ideal of the group ring \(\mathbb{Z}[G]\) (where G is the Galois group), called Stickelberger’s ideal, and show that all its elements annihilate the class group. The proof relies on properties of Gauss sums, an arithmetical object interesting by itself. We develop the theory of Gauss sums to the extent needed for the proof of Stickelberger’s theorem. In the final sections we provide deeper insight into the structure of Stickelberger’s ideal. We determine its \(\mathbb{Z}\)-rank, find a free \(\mathbb{Z}\)-basis, study its real and relative parts, and prove Iwasawa’s class number formula.

In this chapter we continue our study of Mihailescu’s ideal. As follows from the definition, it contains the ideal \(q\mathbb{Z}[G]\) of the elements divisible by q. A basic question is whether Mihailescu’s ideal has nontrivial (that is, not divisible by q) elements.In this chapter we prove that (for large x) the real part \(\mathcal{I}_{M}^{+}\) of Mihailescu’s ideal contains no nontrivial elements of weight 0 and even of any weight divisible by q.On the other hand, later we shall see that a solution to Catalan’s equation gives rise to a nontrivial element of \(\mathcal{I}_{M}^{+}\) of weight divisible by q. This contradiction would prove Catalan’s conjecture.

This chapter is somewhat isolated and can be read (almost) independently of the others. We discuss in it the application of Baker’s method to Diophantine equations of Catalan type. We give a brief introduction to this method, show how it applies to classical Diophantine equations, and reproduce the beautiful argument of Tijdeman, who proved that Catalan’s equation has only finitely many solutions. Moreover, the solutions are bounded by an absolute effective constant (that is, a constant that can, in principle, be explicitly determined), which reduces the problem to a finite computation. Before the work of Mihailescu this was the top achievement on Catalan’s problem.We also consider the more general equation of Pillai and show that it has finitely many solutions when one of the four variables is fixed.

International audience

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.

In this work, we study the diophantine equation \renewcommand{\theequation}{1.1} \begin{equation} f(X) = Y^m, \end{equation} where $m \geqslant 2$ is an integer and $f(X)$ is a polynomial with coefficients in a number field ${\bf K}$ . The first important result on this topic is due to Siegel [19], who showed that if $m = 2$ and $f$ has at least three simple roots or if $m \geqslant 3$ and $f$ has at least two simple roots, then (1.1) has only finitely many integral solutions. Three years later, he proved [20] that if the algebraic curve defined by (1.1) is of positive genus, then (1.1) has only finitely many integral solutions. The $p$ -adic analogue of this theorem was established independently by Lang [9] and LeVeque [12], who showed that, under the same conditions, (1.1) has only finitely many $S$ -integral solutions. After that, LeVeque [13] gave a necessary and sufficient condition for the algebraic curve defined by (1.1) to have positive genus. However, all these results are based on Thue's method, and hence are ineffective.

In this survey chapter, we explain how the theory of Hausdorff dimension and Hausdorff measure is used to answer certain questions in Diophantine approximation. The final section is devoted to a discussion around the Diophantine properties of the points lying in the middle third Cantor set.

Let \(\theta = [0;a_{1},a_{2},\ldots ]\) be an algebraic number of degree at least three. Recently, we have established that the sequence of partial quotients \((a_{\ell})_{\ell\geq 1}\) of θ is not too simple and cannot be generated by a finite automaton. In this expository paper, we point out the main ingredients of the proof and we briefly survey earlier works.

This book presents state-of-the-art research on the distribution modulo one of sequences of integral powers of real numbers and related topics. Most of the results have never before appeared in one book and many of them were proved only during the last decade. Topics covered include the distribution modulo one of the integral powers of 3/2 and the frequency of occurrence of each digit in the decimal expansion of the square root of two. The author takes a point of view from combinatorics on words and introduces a variety of techniques, including explicit constructions of normal numbers, Schmidt's games, Riesz product measures and transcendence results. With numerous exercises, the book is ideal for graduate courses on Diophantine approximation or as an introduction to distribution modulo one for non-experts. Specialists will appreciate the inclusion of over 50 open problems and the rich and comprehensive bibliography of over 700 references.

International audience

Letpbe a prime number. For a positive integernand ap-adic number ξ, let λn(ξ) denote the supremum of the real numbers λ such that there are arbitrarily large positive integersqsuch that ‖qξ‖p,‖qξ2‖p,…,‖qξn‖pare all less thanq−λ−1. Here, ‖x‖pdenotes the infimum of |x−n|pasnruns through the integers. We study the set of values taken by the function λn.

We establish new combinatorial transcendence criteria for continued fraction expansions. Let $\alpha = [0; a_1, a_2,...]$ 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 $\alpha$ cannot be generated by a finite automaton, and that the complexity function of $(a_{\ell})_{\ell \ge 1}$ cannot increase too slowly., Comment: 20 pages; the title has changed

International audience

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)., 8 pages; revised version; to appear in Bull. Lond. Math. Soc

A proof of the transcendence of a real number ! based on the Thue‐Siegel‐Roth‐Schmidt method involves generally a sequence (" n)n! 1 of algebraic numbers of bounded degree or a sequence (xn)n! 1 of integer r-tuples. In the present paper, we show how such a proof can produce a transcendence measure for ! , if one is able to quantify the growth of the heights of the algebraic numbers " n or of the points xn. Our method rests on the quantitative Schmidt subspace theorem. We further give several applications, including to certain normal numbers and to the extremal numbers introduced by Roy.

We prove that there exist arbitrarily small positive real numbers $\epsilon$ such that every integral power $(1 + \vepsilon)^n$ is at a distance greater than $2^{-17} \epsilon |\log \vepsilon|^{-1}$ to the set of rational integers. This is sharp up to the factor $2^{-17} |\log \epsilon|^{-1}$. We also establish that the set of real numbers $\alpha > 1$ such that the sequence of fractional parts $(\{\alpha^n\})_{n \ge 1}$ is not dense modulo 1 has full Hausdorff dimension., Comment: 12 pages

We establish that the set of pairs $(\alpha, \beta)$ of real numbers such that $$ \liminf_{q \to + \infty} q \cdot (\log q)^2 \cdot \Vert q \alpha \Vert \cdot \Vert q \beta \Vert > 0, $$ 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, Comment: 14 pages

Let n be a positive integer. Let ξ be an algebraic real number of degree greater than n. It follows from a deep result of W. M. Schmidt that, for every positive real number ε, there are infinitely many algebraic numbers α of degree at most n such that |ξ - α| < H(α)-n - 1 + ε, where H(α) denotes the naive height of α. We sharpen this result by replacing ε by a function H ε(H) that tends to zero when H tends to infinity. We make a similar improvement for the approximation to algebraic numbers by algebraic integers, as well as for an inhomogeneous approximation problem.

We are studying the Diophantine exponent \mu_{n,l}$ defined for integers 1 \leq l < n and a vector \alpha \in \mathbb{R}^n by letting \mu_{n,l} = \sup{\mu \geq 0: 0 < ||x \cdot \alpha|| < H(x)^{-\mu} 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 \alpha. We calculate the Hausdorff dimension of the set of vectors \alpha with \mu_{n,l} (\alpha) = \mu for \mu \geq n. Finally, letting w_n denote the exponent obtained by removing the restrictions on x, we show that there are vectors \alpha for which the gaps in the increasing sequence \mu_{n,1} (\alpha) \leq ... \leq \mu_{n,n-1} (\alpha) \leq w_n (\alpha) can be chosen to be arbitrary., Comment: 20 pages

A linear form in logarithms of algebraic numbers is an expression of the form $$ \beta _1 \log \alpha _1 + \cdots + \beta _n log \alpha _n , $$ where the α’s and the β’s denote complex algebraic numbers, and log denotes any determination of the logarithm.

Let Θ be a point in Rn. We are concerned with the approximation to Θ by rational linear subvarieties of dimension d for 0 ≤ d ≤ n−1. To that purpose, we introduce various convex bodies in the Grassmann algebra Λ(Rn+1). It turns out that our convex bodies in degree d are the dth compound, in the sense of Mahler, of convex bodies in degree one. A dual formulation is also given. This approach enables us both to split and to refine the classical Khintchine transference principle.

Let C : Y-2 = a(n)X(n)+...+a(0) be a hyperelliptic curve with the ai rational integers, n >= 5, and the polynomial on the right- hand side irreducible. Let J be its Jacobian. We give a completely explicit upper bound for the integral points on the model C, provided we know at least one rational point on C and a Mordell Weil basis for J (Q). We also explain a powerful refinement of the Mordell - Weil sieve which, combined with the upper bound, is capable of determining all the integral points. Our method is illustrated by determining the integral points on the genus 2 hyperelliptic models Y-2 - Y = X-5 - X and (Y 2) = (X 5).

In a recent paper, de Mathan and Teulié asked whether lim infq→+∞q⋅‖qα‖⋅|q|p = 0 holds for every badly approximable real number α and every prime number p. After a survey of the known results on this open problem, we study the analogous question in fields of power series

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)., 31 pages

In various problems in signal theory, the following family of functions of [0, 1]into itself arises naturally : $$ T_{\gamma ,\alpha } :x \mapsto \gamma x + \alpha modulo 1, with 0 < \gamma < 1,0 \leqslant \alpha < 1. $$ The purpose of the present work is to describe very precisely the asymptotic behaviour of the iterates of the functions Tγ,α. Further, we give a continued fraction type algorithm which allows us to obtain, for a given pair (γ,α), the numerical information on the dynamic of Tγ,α.

Suppose that {U-n}(n >= 0) is a Lucas sequence, and suppose that l(1), ..., l(t) are primes. We show that the equation U-n1 ... U-nm =+/- l(1)(x1) ... l(t)(x1)y(p), p prime, m = 0) is the Fibonacci sequence, then we solve the equation F-n1 ... F-nm = 2(x1). 3(x2) . 5(x3) ... 541(x100)y(p) under the restrictions: p is prime and m < p.

International audience; On montre que cette conjecture est vérifiée pour les nombres réels admettant une grande période dans le développement en fraction continue.