On the sum of dilates in Rᵈ

Abstract

Let \(A\) be a nonempty finite subset of \(\mathbb{Z}^d\) which is not contained in a hyperplane, \(q\in\mathbb{Z}\) with \(|q|>1\) and \(m\in \mathbb{Z}\) such that \(|q|+2d-1\leq m\leq (|q|+2d-1)^2\). In this paper it is shown that \[ |A+q\cdot A|\geq \left(\frac{m}{|q|+2d-1}\right)|A|-c \] where \(c\) depends only on \(q,d\) and \(m\). In particular, taking \(m=(|q|+2d-1)^2\), this results confirms a conjecture of A. Balog and G. Shakan.