arXiv subjects

Stéphane Fischler

Publications and source records attributed to Stéphane Fischler.

31 records · Page 2Linked to original sources

arXiv2013

Distribution of irrational zeta values

In this paper we refine Ball-Rivoal's theorem by proving that for any odd integer $a$ sufficiently large in terms of $ε>0$, there exist $[ \frac{(1-ε)\log a}{1+\log 2}]$ odd integers $s$ between 3 and $a$, with distance at least $a^ε$ from one another, at which Riemann zeta function takes $\Q$-linearly independent values. As a consequence, if there are very few integers $s$ such that $ζ(s)$ is irrational, then they are rather evenly distributed. The proof involves series of hypergeometric type estimated by the saddle point method, and the generalization to vectors of Nesterenko's linear independence criterion.

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arXiv2013

Nesterenko's linear independence criterion for vectors

In this paper we deduce a lower bound for the rank of a family of $p$ vectors in $\R^k$ (considered as a vector space over the rationals) from the existence of a sequence of linear forms on $\R^p$, with integer coefficients, which are small at $k$ points. This is a generalization to vectors of Nesterenko's linear independence criterion (which corresponds to $k=1$), used by Ball-Rivoal to prove that infinitely many values of Riemann zeta function at odd integers are irrational. The proof is based on geometry of numbers, namely Minkowski's theorem on convex bodies.

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arXiv2013

A converse to linear independence criteria, valid almost everywhere

We prove a weighted analogue of the Khintchine-Groshev Theorem, where the distance to the nearest integer is replaced by the absolute value. This is subsequently applied to proving the optimality of several linear independence criteria over the field of rational numbers.

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arXiv2012

Connecting Interpolation and Multiplicity Estimates in Commutative Algebraic Groups

Let $G$ be a commutative algebraic group embedded in projective space and $Γ$ a finitely generated subgroup of $G$. From these data we construct a chain of algebraic subgroups of $G$ which is intimately related to obstructions to multiplicity or interpolation estimates. Let $γ_1,...,γ_l$ denote a family of generators of $Γ$ and, for any $S>1$, let $Γ(S)$ be the set of elements $n_1γ_1+..+n_lγ_l$ with integers $n_j$ such that $|n_j| < S$. Then this chain of subgroups controls, for large values of $S$, the distribution of $Γ(S)$ with respect to algebraic subgroups of $G$. As an application we essentially determine (up to multiplicative constants) the locus of common zeros of all $P \in H^0(\barG ,{\cal O}(D))$ which vanish to at least some given order at all points of $Γ(S)$. When $D$ is very small this result reduces to a multiplicity estimate; when $D$ is very large it is a kind of interpolation estimate.

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arXiv2012

Seshadri Constants and Interpolation on Commutative Algebraic Groups

In this article we study interpolation estimates on a special class of compactifications of commutative algebraic groups constructed by Serre. We obtain a large quantitative improvement over previous results due to Masser and the first author and our main result has the same level of accuracy as the best known multiplicity estimates. The improvements come both from using special properties of the compactifications which we consider and from a different approach based upon Seshadri constants and vanishing theorems.

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arXiv2012

Nesterenko's criterion when the small linear forms oscillate

In this paper we generalize Nesterenko's criterion to the case where the small linear forms have an oscillating behaviour (for instance given by the saddle point method). This criterion provides both a lower bound for the dimension of the vector space spanned over the rationals by a family of real numbers, and a measure of simultaneous approximation to these numbers (namely, an upper bound for the irrationality exponent if 1 and only one other number are involved). As an application, we prove an explicit measure of simultaneous approximation to $ζ(5)$, $ζ(7)$, $ζ(9)$, and $ζ(11)$, using Zudilin's proof that at least one of these numbers is irrational.

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arXiv2011

On the values of G-functions

Let f be a G-function (in the sense of Siegel), and x be an algebraic number; assume that the value f(x) is a real number. As a special case of a more general result, we show that f(x) can be written as g(1), where g is a G-function with rational coefficients and arbitrarily large radius of convergence. As an application, we prove that quotients of such values are exactly the numbers which can be written as limits of sequences a(n)/b(n), where the generating series of both sequences are G-functions with rational coefficients. This result provides a general setting for irrationality proofs in the style of Apery for zeta(3), and gives answers to questions asked by T. Rivoal in [Approximations rationnelles des valeurs de la fonction Gamma aux rationnels : le cas des puissances, Acta Arith. 142 (2010), no. 4, 347-365].

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arXiv2009

Irrationality exponent and rational approximations with prescribed growth

Let $ξ$ be a real irrational number. We are interested in sequences of linear forms in 1 and $ξ$, with integer coefficients, which tend to 0. Does such a sequence exist such that the linear forms are small (with given rate of decrease) and the coefficients have some given rate of growth? If these rates are essentially geometric, a necessary condition for such a sequence to exist is that the linear forms are not too small, a condition which can be expressed precisely using the irrationality exponent of $ξ$. We prove that this condition is actually sufficient, even for arbitrary rates of growth and decrease. We also make some remarks and ask some questions about multivariate generalizations connected to Fischler-Zudilin's new proof of Nesterenko's linear independence criterion.

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arXiv2007

Multiple series connected to Hoffman's conjecture on multiple zeta values

Recent results of Zlobin and Cresson-Fischler-Rivoal allow one to decompose any suitable $p$-uple series of hypergeometric type into a linear combination (over the rationals) of multiple zeta values of depth at most $p$; in some cases, only the multiple zeta values with 2's and 3's are involved (as in Hoffman's conjecture). In this text, we study the depth $p$ part of this linear combination, namely the contribution of the multiple zeta values of depth exactly $p$. We prove that it satisfies some symmetry property as soon as the $p$-uple series does, and make some conjectures on the depth $p-1$ part of the linear combination when $p=3$. Our result generalizes the property that (very) well-poised univariate hypergeometric series involve only zeta values of a given parity, which is crucial in the proof by Rivoal and Ball-Rivoal that $ζ(2n+1)$ is irrational for infinitely many $n \geq 1$.

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arXiv2006

Palindromic Prefixes and Diophantine Approximation

This text is devoted to simultaneous approximation to $ξ$ and $ξ^2$ by rational numbers with the same denominator, where $ξ$ is a non-quadratic real number. We focus on an exponent $β_0(ξ)$ that measures the quality of such approximations (when they are exceptionally good). We prove that $β_0$ takes the same set of values as a combinatorial quantity that measures the abundance of palindrome prefixes in an infinite word $w$. This allows us to give a precise exposition of Roy's palindrome prefix method. The main tools we use are Davenport-Schmidt's sequence of minimal points and Roy's bracket operation.

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arXiv2005

Palindromic Prefixes and Episturmian Words

Let $w$ be an infinite word on an alphabet $A$. We denote by $(n_i)_{i \geq 1}$ the increasing sequence (assumed to be infinite) of all lengths of palindrome prefixes of $w$. In this text, we give an explicit construction of all words $w$ such that $n_{i+1} \leq 2 n_i + 1$ for any $i$, and study these words. Special examples include characteristic Sturmian words, and more generally standard episturmian words. As an application, we study the values taken by the quantity $\limsup n_{i+1}/n_i$, and prove that it is minimal (among all non-periodic words) for the Fibonacci word.

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arXiv2003

Irrationalité de valeurs de zêta (d'après Apéry, Rivoal, ...)

This survey text deals with irrationality, and linear independence over the rationals, of values at positive odd integers of Riemann zeta function. The first section gives all known proofs (and connections between them) of Apéry's Theorem (1978) : $ζ(3)$ is irrational. The second section is devoted to a variant of the proof, published by Rivoal and Ball-Rivoal, that infinitely many $ζ(2n+1)$ are irrational. The end of this text deals with more quantitative statements.

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arXiv2002

Formes linéaires en polyzêtas et intégrales multiples

The problem we consider is to define families of n-dimensional integrals, endowed with group actions as in Rhin-Viola's work on irrationality measures of $ζ(2)$ and $ζ(3)$, the values of which are linear forms, over the rationals, in multiple zeta values of weight at most n. We generalize Vasilyev's and Sorokin's approaches, and give a change of variables that connects them to each other. We describe a group structure for a n-dimensional integral that specializes, for n=2 and n=3, to the ones obtained by Rhin and Viola.

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