Sums of Dilates in p

Abstract

We consider the problem of sums of dilates in groups of prime order. It is well known that sets with small density and small sumset in p behave like integer sets. Thus, given A ⊂ p of sufficiently small density, it is straightforward to show that \begin{linenomath} $$| \lambda_{1}A+\lambda_{2}A+\cdots+ \lambda_{k}A | \ge\biggl(\sum_{i}|\lambda_{i}|\biggr)|A|- o(|A|).$$ \end{linenomath} On the other hand, the behaviour for sets of large density turns out to be rather surprising. Indeed, for any ε > 0, we construct subsets of density 1/2–ε such that |A + λ A| ≤ (1–δ)p, showing that there is a very different behaviour for subsets of large density.