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The research demonstrates improved upper bounds on difference set size in abelian groups, highlighting connections with cyclotomic polynomials.
Given a finite abelian group G and a subset J⊂ G J ⊂ G with 0∈ J 0 ∈ J , let DG(J,N) D G ( J , N ) be the maximum size of A⊂ GN A ⊂ G N such that the difference set $$A-A$$ A - A and JN 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 DG(J, N) D G ( J , N ) can be improved exponentially.
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Xu et al. (2026) studied this question.
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