In this work, we show how number theoretical problems can be fruitfully approached with the tools of statistical physics. We focus on g-Sidon sets, which describe sequences of integers whose pairwise sums are different, and propose a random decision problem which addresses the probability of a random set of k integers to be g-Sidon. First, we provide numerical evidence showing that there is a c...

A subspace of a finite extension field is called a Sidon space if the product of any two of its elements is unique up to a scalarmultiplier from the base field. Sidon spaces were recently introduced by Bachoc et al. as a means to characterize multiplicativeproperties of subspaces, and yet no explicit constructions were given. In this paper, several constructions of Sidon spaces arep...

Erdős conjectured the existence of an infinite Sidon sequence of positive integers which is an asymptotic basis of order 3. We progress towards this conjecture in several directions. We prove the conjecture for all cyclic groups ZN with N large enough. We also show that there is an infinite B2[2] sequence which is an asymptotic basis of order 3. Finally, we prove that for any ε > 0 there is a S...

We give the first explicit construction of an infinite Sidon sequence A with A(x) x for some c > 1/3. Our method allows to take c = √ 2− 1. Ruzsa proved the existence of a Sidon sequence with similar counting function but his proof was not constructive. Our method generalizes to Bh sequences when h ≥ 3. In these case our constructions are not explicit but they give the first improvements on the...