Intersective sets over abelian groups
Zixiang Xu, Chi Hoi Yip
TL;DR
This work studies the extremal function $D_G(J,N)$, the maximal size of $A\subset G^N$ with $(A-A)\cap J^N=\{\mathbf{0}\}$ for a finite abelian group $G$ and a subset $J\subset G$ containing $0$. The authors build a bridge between this combinatorial problem and cyclotomic polynomials by encoding $J$-supported polynomials $h$ with $h(0)=1$ and employing eigenvalue methods on weighted Cayley graphs to derive upper bounds of the form $D_G(J,N)\le [G:H]^N\cdot (n-\deg(h))^N$ when $h| (t^n-1)$, with $H=\langle J\rangle$. They further translate cyclotomic structure into effective constructions (via products of cyclotomic polynomials) that yield exponential improvements over the classical $(|G|-|J|+1)^N$ bounds in many cases, and provide matching lower bounds (up to constants) for sets of the form $\{0,a\}$, including explicit corollaries for finite fields. The results extend and sharpen prior bounds by Alon, Heged\H{u}s, and HKP, generalizing to all finite abelian groups, and reveal a deep link between combinatorial difference-avoidance problems and cyclotomic polynomial structure with implications for intersective sets and Cayley-graph spectra.
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 Hegedűs 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.