What question did this study set out to answer?

To generalize and improve known results related to intersective sets in finite abelian groups.

January 6, 2026Designs Codes and Cryptography0 citationsOpen Access

Intersective sets over abelian groups

ZXZixiang Xu CYChi Hoi YipGeorgia Institute of Technology

Key Points

To generalize and improve known results related to intersective sets in finite abelian groups.
Constructed a bridge between extremal problems and cyclotomic polynomials using algebraic graph theory.
Studied the maximum size of difference sets in various groups and subsets.
Identified infinitely many families of G and J where known upper bounds on difference sets can be improved exponentially.

Abstract

Abstract Given a finite abelian group G and a subset J G J ⊂ G with 0 J 0 ∈ J, let D₆ (J, N) D G (J, N) be the maximum size of A G^N A ⊂ G N such that the difference set A-A A - A and J^N 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 D₆ (J, N) D G (J, N) can be improved exponentially.

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