New upper bound for sums of dilates (Q2401423)

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