Group actions and geometric combinatorics in Fd/q

Michael Bennett, Derrick Hart, Alex Iosevich, Jonathan Pakianathan, Misha Rudnev

Research output: Contribution to journalArticle (Academic Journal)peer-review

15 Citations (Scopus)
226 Downloads (Pure)

Abstract

In this paper we apply a group action approach to the study of Erdős–Falconer-type problems in vector spaces over finite fields and use it to obtain non-trivial exponents for the distribution of simplices. We prove that there exists s0(d)<d such that if E⊂Fdq, d≥2, with |E|≥Cqs0, then |Tdd(E)|≥C′q(d+12), where Tdk(E) denotes the set of congruence classes of k-dimensional simplices determined by k+1-tuples of points from E. Non-trivial exponents were previously obtained by Chapman, Erdogan, Hart, Iosevich and Koh [4] for Tdk(E) with 2≤k≤d−1. A non-trivial result for T22(E) in the plane was obtained by Bennett, Iosevich and Pakianathan [2]. These results are significantly generalized and improved in this paper. In particular, we establish the Wolff exponent 43, previously established in [4] for the q≡3mod4 case to the case q≡1mod4, and this results in a new sum-product type inequality. We also obtain non-trivial results for subsets of the sphere in Fdq, where previous methods have yielded nothing. The key to our approach is a group action perspective which quickly leads to natural and effective formulae in the style of the classical Mattila integral from geometric measure theory.
Original languageEnglish
Pages (from-to)91-110
Number of pages20
JournalForum Mathematicum
Volume29
Issue number1
Early online date5 Aug 2016
DOIs
Publication statusPublished - Jan 2017

Keywords

  • Erdős–Falconer problems
  • simplices
  • group action

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