New Upper Bound for Sums of Dilates

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 $\lambda_1, \ldots, \lambda_h \le 2^r$, then $$|\lambda_1 \cdot A + \ldots + \lambda_h \cdot A | \le K^{ 7 rh /\ln (r+h) } |A|.$$ This improves upon a result of Bukh who shows that $$|\lambda_1 \cdot A + \ldots + \lambda_h \cdot A | \le K^{O(rh)} |A|.$$ 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.