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Dimensional lower bounds for Falconer type incidence theorems

Journal d'Analyse Mathématique volume 139, pages 143–154 (2019)Cite this article

Abstract

Let 1 ≤ k ≤ d and consider a subset E ⊂ ℝd. In this paper, we study the problem of how large the Hausdorff dimension of E must be in order for the set of distinct noncongruent k-simplices in E (that is, noncongruent point configurations of k + 1 points from E) to have positive Lebesgue measure. This generalizes the k = 1 case, the well-known Falconer distance problem and a major open problem in geometric measure theory. Many results on Falconer type theorems have been established through incidence theorems, which generally establish sufficient but not necessary conditions for the point configuration theorems. We establish a dimensional lower threshold of \(\frac{d+1}{2}\) on incidence theorems for k-simplices where k ≤ d ≤ 2k + 1 by generalizing an example of Mattila. We also prove a dimensional lower threshold of \(\frac{d+1}{2}\) on incidence theorems for triangles in a convex setting in every dimension greater than 3. This last result generalizes work by Iosevich and Senger on distances that was built on a construction by Valtr. The final result utilizes number-theoretic machinery to estimate the number of solutions to a Diophantine equation.

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Author notes

  1. Jonathan DeWitt

    Present address: Current address Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

  2. Gwyneth Moreland

    Present address: Current address Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

  3. Eyvindur A. Palsson

    Present address: Current address Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

Authors and Affiliations

  1. Department of Mathematics & Statistics, Haverford College, Haverford, PA, 19041, USA

    Jonathan DeWitt

  2. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Kevin Ford

  3. Department of Mathematics & Statistics, Williams College, Williamstown, MA, 01267, USA

    Eli Goldstein, Steven J. Miller & Eyvindur A. Palsson

  4. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Gwyneth Moreland

  5. Department of Mathematics, Missouri State University, Springfield, MO, 65897, USA

    Steven Senger

Authors
  1. Jonathan DeWitt
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  2. Kevin Ford
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  3. Eli Goldstein
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  4. Steven J. Miller
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  5. Gwyneth Moreland
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  6. Eyvindur A. Palsson
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  7. Steven Senger
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Corresponding author

Correspondence to Eyvindur A. Palsson.

Additional information

The first, third, fourth and fifth authors were supported in part by National Science Foundation grants DMS1265673, DMS1561945 and DMS1347804. The second-listed author was supported in part by National Science Foundation grant DMS1501982 and the sixth-listed author was supported supported in part by Simons Foundation Grant #360560. The authors thank an anonymous referee for suggestions that significantly improved the paper.

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DeWitt, J., Ford, K., Goldstein, E. et al. Dimensional lower bounds for Falconer type incidence theorems. JAMA 139, 143–154 (2019). https://doi.org/10.1007/s11854-019-0056-0

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  • DOI: https://doi.org/10.1007/s11854-019-0056-0