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.
@article{10_37236_6576,
author = {Albert Bush and Yi Zhao},
title = {New upper bound for sums of dilates},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {3},
doi = {10.37236/6576},
zbl = {1407.11118},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6576/}
}
TY - JOUR
AU - Albert Bush
AU - Yi Zhao
TI - New upper bound for sums of dilates
JO - The electronic journal of combinatorics
PY - 2017
VL - 24
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/6576/
DO - 10.37236/6576
ID - 10_37236_6576
ER -
%0 Journal Article
%A Albert Bush
%A Yi Zhao
%T New upper bound for sums of dilates
%J The electronic journal of combinatorics
%D 2017
%V 24
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/6576/
%R 10.37236/6576
%F 10_37236_6576
Albert Bush; Yi Zhao. New upper bound for sums of dilates. The electronic journal of combinatorics, Tome 24 (2017) no. 3. doi: 10.37236/6576
