Sum of dilates in vector spaces

TL;DR

This paper establishes a lower bound on the size of the sumset of a finite subset of integer lattice points and its dilate, revealing new insights into the additive structure of such sets in higher dimensions.

Contribution

It provides a new lower bound for the sum of a set and its dilate in vector spaces, extending previous results to higher dimensions with explicit bounds.

Findings

Established a lower bound for |A + q·A| in initely generated integer lattice sets.

Extended additive combinatorics results to higher dimensions.

Demonstrated the bound's dependence on dimension and dilation factor.

Abstract

Let $d\geq 2$, $A \subset \mathbb{Z}^d$ be finite and not contained in a translate of any hyperplane, and $q \in \mathbb{Z}$ such that $|q| > 1$. We show $$|A+ q \cdot A| \geq (|q|+d+1)|A| - O_{q,d}(1).$$