Additive Volume of Sets Contained in Few Arithmetic Progressions

TL;DR

This paper extends Freiman's conjecture on the maximum volume of finite integer sets with given size and doubling, proving the formula for sets of three segments and exploring structural properties.

Contribution

It generalizes Freiman's volume formula to higher dimensions and specific structured sets, including those made of three segments.

Findings

Proves the volume formula for sets of three segments.

Provides structural insights into extremal sets.

Discusses extensions to sets with bounded segments.

Abstract

A conjecture of Freiman gives an exact formula for the largest volume of a finite set $A$ of integers with given cardinality $k = |A|$ and doubling $T = |2A|$. The formula is known to hold when $T \le 3k-4$, for some small range over $3k-4$ and for families of structured sets called chains. In this paper we extend the formula to sets of every dimension and prove it for sets composed of three segments, giving structural results for the extremal case. A weaker extension to sets composed of a bounded number of segments is also discussed.