New lower bounds for cardinalities of higher dimensional difference sets and sumsets
Abstract
Let $d \geq 4$ be a natural number and let $A$ be a finite, nonempty subset of $\mathbb{R}^d$ such that $A$ is not contained in a translate of a hyperplane. In this setting, we show that \[ AA \geq \bigg(2d  2 + \frac{1}{d1} \bigg) A  O_{d}(A^{1 \delta}), \] for some absolute constant $\delta>0$ that only depends on $d$. This provides a sharp main term, consequently answering questions of Ruzsa and Stanchescu up to an $O_{d}(A^{1 \delta})$ error term. We also prove new lower bounds for restricted type difference sets and asymmetric sumsets in $\mathbb{R}^d$.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.11300
 Bibcode:
 2021arXiv211011300M
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Number Theory;
 11B13;
 11B30
 EPrint:
 18 pages