Given a finite abelian group G and a subset <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$J\subset G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>J</mi> <mo>⊂</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$0\in J$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>∈</mo> <mi>J</mi> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$D_{G}(J,N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>J</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the maximum size of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A\subset G^{N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>⊂</mo> <msup> <mi>G</mi> <mi>N</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> such that the difference set <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$A-A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>-</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$J^{N}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>J</mi> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$D_{G}(J, N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>J</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> can be improved exponentially.

Intersective sets over abelian groups

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2026年1月5日

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Zixiang Xu,
Chi Hoi Yip

摘要

Given a finite abelian group G and a subset $J\subset G$ $J \subset G$ with $0\in J$ $0 \in J$ , let $D_{G}(J,N)$ $D_{G} (J, N)$ be the maximum size of $A\subset G^{N}$ $A \subset G^{N}$ 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 $D_{G}(J, N)$ $D_{G} (J, N)$ can be improved exponentially.