Upper Bounds on the Cardinality of Higher Sumsets

TL;DR

This paper establishes improved upper bounds on the size of higher sumsets in finite sets within commutative groups, refining previous bounds by incorporating a decreasing factor related to h.

Contribution

It introduces a submultiplicative upper bound on |A+hB| that improves upon Ruzsa's earlier bounds by including a diminishing factor as h increases.

Findings

Provides a tighter upper bound on sumset sizes.

Introduces a submultiplicative bound decreasing with h.

Enhances understanding of sumset growth in additive combinatorics.

Abstract

Let A and B be finite sets in a commutative group. We bound |A+hB| in terms of |A|, |A+B| and h. We provide a submultiplicative upper bound that improves on the existing bound of Imre Ruzsa by inserting a factor that decreases with h.