A generalization of Croot-Lev -Pach's Lemma and a new upper bound for the size of difference sets in polynomial rings

Gábor Hegedüs

Published 2018-03-14Version 1

Croot, Lev and Pach used a new polynomial technique to give a new exponential upper bound for the size of three-term progression-free subsets in the groups $(\mathbb Z_4)^n$. The main tool in proving their breakthroughing result is a simple lemma about polynomials, which gives interesting new bounds for the size of subsets of the vector space $({\mathbb Z_p})^n$. Our main result is a generalization of this lemma. In the proof we combined Tao's slice rank bounding method with Gr\"obner basis technique. As an application, we improve Green's results and present new upper bounds for the size of difference sets in polynomial rings. We give a new, more concrete upper bound for the size of arithmetic progression-free subsets in $({\mathbb Z_p})^n$.