Improved bounds for arithmetic progressions in product sets

Abstract

Let B be a set of natural numbers of size n. We prove that the length of the longest arithmetic progression contained in the product set B.B = {bb′|b, b′ ∈ B} cannot be greater than O(n log n) which matches the lower bound provided in an earlier paper up to a multiplicative constant. For sets of complex numbers, we improve the bound to O ϵ (n 1 + ϵ ) for arbitrary ϵ > 0 assuming the GRH.

Locations

• International Journal of Number Theory - View
• arXiv (Cornell University) - View - PDF