Complete $3$-term arithmetic progression free sets of small size in vector spaces and other abelian groups

Bence Csajbók; Zoltán Lóránt Nagy

Complete $3$-term arithmetic progression free sets of small size in vector spaces and other abelian groups

Combinatorics 2025-09-30 v4 Number Theory

Authors: Bence Csajbók , Zoltán Lóránt Nagy

Abstract

A subset $S$ of an abelian group $G$ is called $3$ - $\mathrm{AP}$ free if it does not contain a three term arithmetic progression. Moreover, $S$ is called complete $3$ - $\mathrm{AP}$ free, if it is maximal w.r.t. set inclusion. One of the most central problems in additive combinatorics is to determine the maximal size of a $3$ - $\mathrm{AP}$ free set, which is necessarily complete. In this paper we are interested in the minimum size of complete $3$ - $\mathrm{AP}$ free sets. We define and study saturation w.r.t. $3$ - $\mathrm{AP}$ s and present constructions of small complete $3$ - $\mathrm{AP}$ free sets and $3$ - $\mathrm{AP}$ saturating sets for several families of vector spaces and cyclic groups.

Keywords

3-AP free sets complete 3-AP free sets additive combinatorics vector spaces cyclic groups saturating sets

Cite

@article{arxiv.2401.06283,
  title  = {Complete $3$-term arithmetic progression free sets of small size in vector spaces and other abelian groups},
  author = {Bence Csajbók and Zoltán Lóránt Nagy},
  journal= {arXiv preprint arXiv:2401.06283},
  year   = {2025}
}

Comments

Definition 1.8, Proposition 2.3, Theorem 4.14 and Corollary 4.15 are revised

Complete $3$-term arithmetic progression free sets of small size in vector spaces and other abelian groups

Abstract

Keywords

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