Group actions and geometric combinatorics in 𝔽qd$\mathbb{F}_{q}^{d}$

Abstract

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>s</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>d</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>&lt;</m:mo> <m:mi>d</m:mi> </m:mrow> </m:math> ${s_{0}(d)&lt;d}$ such that if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>E</m:mi> <m:mo>⊂</m:mo> <m:msubsup> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> <m:mi>d</m:mi> </m:msubsup> </m:mrow> </m:math> ${E\subset{\mathbb{F}}_{q}^{d}}$ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> ${d\geq 2}$ , with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mi>E</m:mi> <m:mo>|</m:mo> </m:mrow> <m:mo>≥</m:mo> <m:mrow> <m:mi>C</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>q</m:mi> <m:msub> <m:mi>s</m:mi> <m:mn>0</m:mn> </m:msub> </m:msup> </m:mrow> </m:mrow> </m:math> ${|E|\geq Cq^{s_{0}}}$ , then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mrow> <m:msubsup> <m:mi>T</m:mi> <m:mi>d</m:mi> <m:mi>d</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>E</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mo>≥</m:mo> <m:mrow> <m:msup> <m:mi>C</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:msup> <m:mi>q</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mfrac> <m:mrow> <m:mi>d</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mn>2</m:mn> </m:mfrac> <m:mo>)</m:mo> </m:mrow> </m:msup> </m:mrow> </m:mrow> </m:math> ${|T^{d}_{d}(E)|\geq C^{\prime}q^{d+1\choose 2}}$ , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>T</m:mi> <m:mi>k</m:mi> <m:mi>d</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>E</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${T^{d}_{k}(E)}$ denotes the set of congruence classes of k -dimensional simplices determined by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> ${k+1}$ -tuples of points from E . Non-trivial exponents were previously obtained by Chapman, Erdogan, Hart, Iosevich and Koh [4] for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>T</m:mi>