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Finite configurations in sparse sets

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Abstract

Let E ⊆ ℝn be a closed set of Hausdorff dimension α. For m > n, let{B 1, …, B k } be n × (m − n) matrices. We prove that if the system of matrices B j 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 {B 1 y, …, B k y}. 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.

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Authors and Affiliations

  1. University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada, V6T 1Z2

    Vincent Chan, Izabella Łaba & Malabika Pramanik

Authors
  1. Vincent Chan
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  2. Izabella Łaba
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  3. Malabika Pramanik
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Corresponding author

Correspondence to Vincent Chan.

Additional information

The second and third authors were supported by NSERC Discovery Grants.

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Chan, V., Łaba, I. & Pramanik, M. Finite configurations in sparse sets. JAMA 128, 289–335 (2016). https://doi.org/10.1007/s11854-016-0010-3

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  • DOI: https://doi.org/10.1007/s11854-016-0010-3

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