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.