New Upper Bound for Sums of Dilates

Abstract

For $\lambda \in \mathbb{Z}$, let $\lambda \cdot A=\{\lambda a:a\in A\}$. Suppose $r,h\in \mathbb{Z}$ are sufficiently large and comparable to each other. We prove that if $|A+A|\le K|A|$ and ${\lambda }_1,\dots ,{\lambda }_h\le 2^r$, then \[ |λ_1 \cdot A + \ldots + λ_h \cdot A | \le K^{ 7 rh /\ln (r+h) } |A|. \] This improves upon a result of Bukh who shows that \[ |λ_1 \cdot A + \ldots + λ_h \cdot A | \le K^{O(rh)} |A|. \] Our main technique is to combine Bukh's idea of considering the binary expansion of ${\lambda }_i$ with a result on biclique decompositions of bipartite graphs.lique decompositions.