New Upper Bound for Sums of Dilates

Albert Bush; Yi Zhao

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arXiv · 1607.04888

New Upper Bound for Sums of Dilates

Albert Bush·Yi Zhao

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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 $\lambda_1, \ldots, \lambda_h \le 2^r$, 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.

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Albert Bush, Yi Zhao. 2016-07-17. New Upper Bound for Sums of Dilates. https://arxiv.org/abs/1607.04888

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