Discrete spheres and arithmetic progressions in product sets
Volume 178 / 2017
Abstract
We prove that if is a set of N positive integers such that B\cdot B contains an arithmetic progression of length M, then for some absolute C \gt 0, \pi(M) + C \frac {M^{2/3}}{\log^2 M} \leq N, where \pi is the prime counting function. This improves on previously known bounds of the form N = \Omega(\pi(M)) and gives a bound which is sharp up to the second order term, as Pách and Sándor gave an example for which N \lt \pi(M)+ O\biggl(\frac {M^{2/3}}{\log^2 M} \biggr). The main new tool is a reduction of the original problem to the question of approximate additive decomposition of the 3-sphere in \mathbb{F}_3^n which is the set of 0-1 vectors with exactly three non-zero coordinates. Namely, we prove that such a set cannot have an additive basis of order two of size less than c n^2 with absolute constant c \gt 0.