10 April 2014, a general meeting of AMU was held, which was addressed by Prof. Rom Pinchasi from Israel Institute of Technology with a report “On the union of arithmetic progressions”.

Abstract: We show that for every \(\varepsilon>0\) there is an absolute constant \(c(\varepsilon)>0\) such that the following is true: The union of any n arithmetic progressions, each of length n, with pairwise distinct differences must consist of at least \(c(\varepsilon)n^{2-\varepsilon}\) elements. We show also that this type of bound is essentially best possible, as we observe \(n\) arithmetic progressions, each of length n, with pairwise distinct differences such that their cardinality of their union is \(o(n^2)\).

We develop some number theoretical tools that are of independent interest. In particular we give almost tight bounds on the following question: Given \(n\) distinct integers \(a_1,…,a_n\) at most how many pairs satisfy \(a_j/a_i\in [n]\)? More tight bounds on natural related problems will be presented.

This is joint work with Shoni Gilboa.