András Biró

Strong characterizing sequences in simultaneous diophantine approximation

Abstract Answering a question of Liardet, we prove that if 1,α1,α2,…,αt are real numbers linearly independent over the rationals, then there is an infinite subset A of the positive integers such that for real β, we have ( || || denotes the distance to the nearest integer) ∑ n∈A ||nβ|| if and only if β is a linear combination with integer coefficients of 1,α1,α2,…,αt. The proof combines elementary ideas with a deep theorem of Freiman on set addition. Using Freimans theorem, we prove a lemma on the structure of Bohr sets, which may have independent interest.

Number Theory, Analysis, and Combinatorics: Proceedings of the Paul Turan Memorial Conference held August 22-26, 2011 in Budapest

Paul Turan, one of the greatest Hungarian mathematicians, was born 100 years ago, on August 18, 1910. To celebrate this occasion the Hungarian Academy of Sciences, the Alfred Renyi Institute of Mathematics, the Janos Bolyai Mathematical Society and the Mathematical Institute of Eotvos Lorand University organized an international conference devoted to Paul Turans main areas of interest: number theory, selected branches of analysis, and selected branches of combinatorics. The conference was held in Budapest, August 22-26, 2011. Some of the invited lectures reviewed different aspects of Paul Turans work and influence. Most of the lectures allowed participants to report about their own work in the above mentioned areas of mathematics.

Some integrals of hypergeometric functions

We consider a certain definite integral involving the product of two classical hypergeometric functions having complicated arguments. We show the surprising fact that this integral does not depend on the parameters of the hypergeometric functions.

Interpolation by elliptic functions

The starting point of this article is an unpublished result of G. Halász stating that if Ω ⊂ C is a fixed lattice, there are n ≥ 3 given points (s i , 1 ≤ i ≤ n) different modulo Ω and there are given values w i  ∈ C, then there is a function f elliptic with respect to Ω of order at most n − 1 such that f(s i ) = w i for 1 ≤ i ≤ n. We prove that if n ≥ 6, then under the obvious necessary condition there is no 1 ≤ j ≤ n such that w i  = w for every i ≠ j, 1 ≤ i ≤ n, but w j  ≠ w with some w ∈ C, one can improve the upper bound n − 1 for the order of f, i.e. one can give an interpolating f of order at most n − 2.