Finite configurations in sparse sets

By Vincent Chan, I. Łaba, M. Pramanik. 2013

59 citations recorded by the source corpus.

Let E ⊆ ℝn be a closed set of Hausdorff dimension α. For m > n, let{B1, …, Bk} be n × (m − n) matrices. We prove that if the system of matrices Bj is non-degenerate in a suitable sense, α is sufficiently close to n, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then for a range of m depending on n and k, the set E contains a translate of a non-trivial k-point configuration {B1y, …, Bky}. As a consequence, we are able to establish existence of certain geometric configurations in Salem sets (such as parallelograms in ℝn and isosceles right triangles in ℝ2). This can be viewed as a multidimensional analogue of the result of [25] on 3-term arithmetic progressions in subsets of ℝ.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/33f818ed14f65ca1c0f7dc1bc0cd00fabce4fa1c

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In Discrete Mathematics and Combinatorics

  1. Alex Iosevich
  2. Michael A. Bennett
  3. Samir Siksek

In Geometry and Topology

  1. Alex Iosevich
  2. Michael A. Bennett
  3. Andrej Dujella
  4. Samir Siksek
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  6. B. Totaro

In Mathematical Physics

  1. Alex Iosevich
  2. Michael A. Bennett
  3. B. Totaro

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