Abstract
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 ⊂ R, we prove that there exist a, a′ ∈ A such that ∣∣∣∣∣ (aA + 1)(2) (a′ A + 1)(2) (aA + 1)(2) (a′ A + 1) ∣≳|A|31/12. We are also able to prove that max{|A + A − A|, |A2 + A2 − A2 |, |A3 + A3 − A3 |} ≳ |A|19/12. Both of these bounds are improvements of recent results and takes advantage of computer algebra to tackle some of the computations.
| Original language | English |
|---|---|
| Article number | #P1.3 |
| Journal | Electronic Journal of Combinatorics |
| Volume | 31 |
| Issue number | 1 |
| DOIs | |
| State | Published - 2024 |
Funding
E. Wong acknowledges that this manuscript has been partially authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy (DOE). The publisher acknowledges the US government license to provide public access under the DOE Public Access Plan. O. Roche-Newton was supported by the Austrian Science Fund FWF Project P34180. E. Wong was mostly supported by the Austrian Academy of Sciences at the Radon Institute for Computational and Applied Mathematics (RICAM) during the writing and preparation of this article. We are grateful to Orit Raz and Audie Warren for helpful input when putting together this paper.