On the volumes of simplices determined by a subset of R^d

Authors

  • Pablo Shmerkin The University of British Columbia, Department of Mathematics
  • Alexia Yavicoli The University of British Columbia, Department of Mathematics

Keywords:

Patterns, configurations, simplices, volumes, Hausdorff dimension, slices, projections

Abstract

We prove that for 1k<d, if E is a Borel subset of Rd of Hausdorff dimension strictly larger than k, the set of (k+1)-volumes determined by k+2 points in E has positive one-dimensional Lebesgue measure. In the case k=d1, we obtain an essentially sharp lower bound on the dimension of the set of tuples in E generating a given volume. We also establish a finer version of the classical slicing theorem of Marstrand–Mattila in terms of dimension functions, and use it to extend our results to sets of "dimension logarithmically larger than k".
Section
Articles

Published

2025-03-13

How to Cite

Shmerkin, P., & Yavicoli, A. (2025). On the volumes of simplices determined by a subset of R^d. Annales Fennici Mathematici, 50(1), 97–108. https://doi.org/10.54330/afm.159807