This paper explores the relationship between convexity and sum sets. In particular, we show that elementary number theoretical methods, principally the application of a squeezing principle, can be augmented with the Elekes-Szabó Theorem in order to give new information. Namely, if we let $A \subset \mathbb R$, we prove that there exist $a,a' \in A$ such that\[\left | \frac{(aA+1)^{(2)}(a'A+1)^{(2)}}{(aA+1)^{(2)}(a'A+1)} \right | \gtrsim |A|^{31/12}.\]We are also able to prove that\[\max \{|A+ A-A|, |A^2+A^2-A^2|, |A^3 + A^3 - A^3|\} \gtrsim |A|^{19/12}.\]Both of these bounds are improvements of recent results and takes advantage of computer algebra to tackle some of the computations.
@article{10_37236_11331,
author = {Oliver Roche-Newton and Elaine Wong},
title = {Convexity, squeezing, and the {Elekes-Szab\'o} theorem},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {1},
doi = {10.37236/11331},
zbl = {1540.05009},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11331/}
}
TY - JOUR
AU - Oliver Roche-Newton
AU - Elaine Wong
TI - Convexity, squeezing, and the Elekes-Szabó theorem
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/11331/
DO - 10.37236/11331
ID - 10_37236_11331
ER -
%0 Journal Article
%A Oliver Roche-Newton
%A Elaine Wong
%T Convexity, squeezing, and the Elekes-Szabó theorem
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/11331/
%R 10.37236/11331
%F 10_37236_11331
Oliver Roche-Newton; Elaine Wong. Convexity, squeezing, and the Elekes-Szabó theorem. The electronic journal of combinatorics, Tome 31 (2024) no. 1. doi: 10.37236/11331
