Intersective sets over abelian groups

TL;DR

This paper investigates extremal problems in finite abelian groups, improving bounds on the size of subsets avoiding certain difference intersections by connecting algebraic graph theory and cyclotomic polynomials.

Contribution

It generalizes previous results and introduces a novel approach linking algebraic graph theory with cyclotomic polynomials to improve bounds exponentially.

Findings

Constructed infinite families of groups and subsets with improved bounds

Connected extremal problems to cyclotomic polynomials and algebraic graph theory

Achieved exponential improvements over existing upper bounds

Abstract

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. Recently, this extremal problem has been widely studied for different groups $G$ and subsets $J$. In this paper, we generalize and improve the relevant results by Alon and by Heged\H{u}s by building a bridge between this problem and cyclotomic polynomials with the help of algebraic graph theory. In particular, we construct infinitely many non-trivial families of $G$ and $J$ for which the current known upper bounds on $D_{G}(J, N)$ can be improved exponentially.