Sums of dilates over groups of prime order

TL;DR

This paper investigates the size of sums of dilates over prime order groups, establishing near-optimal bounds for fixed density and large dilation factors as the prime tends to infinity.

Contribution

It provides new bounds on the minimal size of sumsets of the form A + λ·A in prime groups for large λ and fixed density, advancing understanding in additive combinatorics.

Findings

Established near-optimal bounds for sumsets with large dilates

Analyzed behavior as prime p tends to infinity for fixed density

Extended previous results to new regimes of λ and density

Abstract

For $p$ prime, $A \subseteq \mathbb{Z}/p\mathbb{Z}$ and $\lambda \in \mathbb{Z}$, the sum of dilates $A + \lambda \cdot A$ is defined by \[A + \lambda \cdot A = \{a + \lambda a' : a, a' \in A\}.\] The basic problem on such sums of dilates asks for the minimum size of $|A + \lambda \cdot A|$ for given $\lambda$, $A$ of given density $\alpha$, and $p$ tending to infinity. We investigate this problem for $\alpha$ fixed and $\lambda$ tending to infinity, proving near-optimal bounds in this case.