Sums of dilates over groups of prime order

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 \[A + λ\cdot A = \{a + λa' : a, a' \in A\}.\] 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.

9 pages. To appear in The American Mathematical Monthly