Sums of Linear Transformations in Higher Dimensions

  title={Sums of Linear Transformations in Higher Dimensions},
  author={Akshat Mudgal},
  journal={The Quarterly Journal of Mathematics},
  • Akshat Mudgal
  • Published 20 February 2019
  • Mathematics
  • The Quarterly Journal of Mathematics
In this paper, we prove the following two results. Let d be a natural number and q, s be co-prime integers such that 10 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… 

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