Towards $3n-4$ in groups of prime order

TL;DR

This paper proves that small subsets of prime order groups with limited doubling are contained within small arithmetic progressions, advancing understanding of additive structure in such groups.

Contribution

It improves existing results by establishing tighter conditions under which subsets of prime order groups are contained in arithmetic progressions.

Findings

Subsets with small doubling are contained in small arithmetic progressions.

The sumset contains a large arithmetic progression with at least 2|A|-1 terms.

Conditions on subset size and doubling constant are refined.

Abstract

We show that if $A$ is a subset of a group of prime order $p$ such that $|2A|<2.7652|A|$ and $|A|<1.25\cdot10^{-6}p$, then $A$ is contained in an arithmetic progression with at most $|2A|-|A|+1$ terms, and $2A$ contains an arithmetic progression with the same difference and at least $2|A|-1$ terms. This improves a number of previously known results.