Convexity, Squeezing, and the Elekes-Szabó Theorem

Journal Article · 11 January 2024 · The Electronic Journal of Combinatorics

DOI: https://doi.org/10.37236/11331 · OSTI ID:2301656

Roche-Newton, Oliver ^[1]; Wong, Elaine ^[2]

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1. Johannes Kepler Univ. Linz (Austria)
2. Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Here, 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.

• Research Organization: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
• Sponsoring Organization: USDOE
• Grant/Contract Number: AC05-00OR22725
• OSTI ID: 2301656
• Journal Information: The Electronic Journal of Combinatorics, Journal Name: The Electronic Journal of Combinatorics Journal Issue: 1 Vol. 31; ISSN 1077-8926
• Publisher: International PressCopyright Statement
• Country of Publication: United States
• Language: English