Sums of Linear Transformations in Higher Dimensions

Abstract

AbstractIn this paper, we prove the following two results. Let d be a natural number and q, s be co-prime integers such that 1<qs<|q|. Then there exists a constant δ>0 depending only on q, s and d such that for any finite subset A of ℝd that is not contained in a translate of a hyperplane, we have |q⋅A+s⋅A|≥(|q|+|s|+2d−2)|A|−Oq,s,d(|A|1−δ).The main term in this bound is sharp and improves upon an earlier result of Balog and Shakan. Secondly, let L∈GL2(ℝ) be a linear transformation such that L does not have any invariant one-dimensional subspace of ℝ2. Then, for all finite subsets A of ℝ2, we have |A+L(A)|≥4|A|−O(|A|1−δ), for some absolute constant δ>0. The main term in this result is sharp as well.