New Upper Bound for Sums of Dilates

TL;DR

This paper establishes a new upper bound for the size of sums of dilates of a set, improving previous bounds by combining binary expansion techniques with biclique decomposition methods.

Contribution

It introduces a novel bound on sums of dilates, enhancing prior results by integrating binary expansion analysis and biclique decomposition strategies.

Findings

New upper bound: |λ₁·A + ... + λ_h·A| ≤ K^{7 rh / ln(r+h)} |A|

Improvement over Bukh's bound of K^{O(rh)}

Technique combines binary expansion with biclique decompositions

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.