Product sets of arithmetic progressions

Abstract: We prove that the size of the product set of any finite arithmetic progression A in integers of size N is at least N^{2}/(log N)^{c+o(1)}, where c=1-(1+loglog 2)/(log 2). This matches the bound in the celebrated Erdos multiplication table problem, up to a factor of (log N)^{o(1)} and thus confirms a conjecture of Elekes and Ruzsa. If instead A is relaxed to be a subset of a finite arithmetic progression in integers with positive constant density, we prove that the size of the product set is at least N^{2}/(log N)^{2log2-1 + o(1)}. This solves the typical case of another conjecture of Elekes and Ruzsa on the size of the product set of a set A whose sum set is of size O(|A|).This is joint work with Yunkun Zhou.

combinatoricsnumber theory

Audience: researchers in the topic

Webinar in Additive Combinatorics

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