Cyclotomic polynomials and intersective sets over abelian groups

Given a finite abelian group $G$ and a subset $J\subset G$ with $0\in J$, let $D_{G}(J,N)$ be the maximum size of $A\subset G^{N}$ such that the difference set $A-A$ and $J^{N}$ have no non-trivial intersection. In this talk, I will describe some recent progress on this S\'ark\"ozy type problem. In particular,  by building a bridge between this problem and cyclotomic polynomials, we obtain generalization and improvement on the recent results by Alon and by Heged\H{u}s. As a consequence, we construct infinitely many non-trivial families of $G$ and $J$ for which the upper bounds on $D_G(J,N)$ obtained by them (via linear algebra method) can be improved exponentially.

This is a joint work with Zixiang Xu.