On the size of sets avoiding a general structure

Runze Wang – GJM, Volume 9, Issue 2 (2024), 22-26.

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janvier 22, 2025
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Given a finite abelian group $G$ and a subset $\subseteq G$ , we let $N_{G,\ S}$ be the smallest integer $N$ such that for any subset $A\subseteq G$ with $N$ elements, we have $g+S\subseteq A$ for some $g\in G$ . Using the probabilistic method, we prove that
$\frac{|H_G(S)|-1}{|H_G(S)|}|G|+\Biggl\lceil\biggl(\frac{|G|}{|H_G(S)|}\biggr)^{1-|H_G(S)|/|S|}\Biggr\rceil\le N_{G,\ S}\le \biggl\lfloor\frac{|S|-1}{|S|}|G|\biggr\rfloor+1$ ,
where $H_G(S)$ is the stabilizer of $S$ .

Categories: Issue2

Milestones:

Received: September 25, 2024
Accepted: December 27, 2024
Revised: December 28, 2024

Authors:

Runze Wang
Department of Mathematical Sciences,
University of Memphis, Memphis, TN 38152, USA.

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