New Upper Bound for Sums of Dilates

For $\lambda \in \mathbb{Z}$, let $\lambda \cdot A = { \lambda a : a \in A}$\mathrm{.}Suppose$r, h\in $\mathbb{Z}$$ are sufficiently large and comparable to each other. We prove that if $\mathrm{\mid }A+A\mathrm{\mid }\le K\mathrm{\mid }A\mathrm{\mid }$ and $\lambda_1, \ldots, \lambda_h \le 2^r$, then $$\mathrm{\mid }{\lambda }_1\cdot A+\dots +{\lambda }_h\cdot A\mathrm{\mid }\le K^{7rh\mathrm{/}\ln (r+h)}\mathrm{\mid }A\mathrm{\mid }\mathrm{.}$$ This improves upon a result of Bukh who shows that $$\mathrm{\mid }{\lambda }_1\cdot A+\dots +{\lambda }_h\cdot A\mathrm{\mid }\le K^{O(rh)}\mathrm{\mid }A\mathrm{\mid }\mathrm{.}$$ 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.