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September 30, 2025

Sumset growth in progression-free sets

Key Points

The growth of sumsets in progression-free sets reveals a lower bound on cardinality, enhancing understanding of combinatorial structures.
It is shown that |A+B|≥|A|²k−1/3k−2|B|k3k−2 in defined groups, establishing significant properties of their structures.
Upper bounds derived for summands in sumsets A+B+C reveal constraints, particularly for sets of t-th powers.
Conditions for minimal cardinality show that min(|A|,|B|,|C|) is limited by logarithmic factors, indicating growth restrictions.

Abstract

We study the growth of sumsets A+B⊂S⊂G, where S does not contain an arithmetic progression of length 2k+1, and where G is a commutative group, in which every nonzero element has an order of at least 2k+1. More specifically, we show the following: if A,B⊂G are sets such that A+B does not contain an arithmetic progression of length 2k+1, then |A+B|≥|A|2k−13k−2|B|k3k−2. As an application we derive upper bounds on the cardinality of the summands in sumsets A+B+C contained in the set of t-th powers, where t≥2 is an integer. In particular, we show that min(|A|,|B|,|C|)≪(logN)4/5 for t=2, and min(|A|,|B|,|C|)≪t(logN)1/2 for t≥3.

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