On the sum of dilates in Rᵈ - Huicochea, Mario (North-Western European Journal of Mathematics, 2021)
Let \(A\) be a nonempty finite subset of \(Z^d\) which is not contained in a hyperplane, \(qZ\) with \(|q|>1\) and \(mZ\) such that \(|q|+2d-1≤ m≤ (|q|+2d-1)^2\). In this paper it is shown that \[ |A+q· A|≥ (m|q|+2d-1)|A|-c \] 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.