In this paper we prove new bounds for sums of convex or concave functions. Specifically, we prove that for all $A,B \subseteq \mathbb R$ finite sets, and for all $f,g$ convex or concave functions, we have $$|A + B|^{38}|f(A) + g(B)|^{38} \gtrsim |A|^{49}|B|^{49}.$$ This result can be used to obtain bounds on a number of two-variable expanders of interest, as well as to the asymmetric sum-product problem. We also adjust our technique to prove the three-variable expansion result $$|AB+A|\gtrsim |A|^{\frac{3}{2} +\frac{3}{170}}\,.$$Our methods follow a series of recent developments in the sum-product literature, presenting a unified picture. Of particular interest is an adaptation of a regularisation technique of Xue, originating in a paper of Rudnev, Shakan, and Shkredov, that enables us to find positive proportion subsets with certain desirable properties.
@article{10_37236_10852,
author = {Sophie Stevens and Audie Warren},
title = {On sum sets and convex functions},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {2},
doi = {10.37236/10852},
zbl = {1495.11021},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10852/}
}
TY - JOUR
AU - Sophie Stevens
AU - Audie Warren
TI - On sum sets and convex functions
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/10852/
DO - 10.37236/10852
ID - 10_37236_10852
ER -
%0 Journal Article
%A Sophie Stevens
%A Audie Warren
%T On sum sets and convex functions
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/10852/
%R 10.37236/10852
%F 10_37236_10852
Sophie Stevens; Audie Warren. On sum sets and convex functions. The electronic journal of combinatorics, Tome 29 (2022) no. 2. doi: 10.37236/10852
