Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces

The first goal of this paper is to prove a sharp condition to guarantee of having a positive proportion of all congruence classes of triangles in given sets in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> <m:mn>2</m:mn> </m:msubsup> </m:math> {Fq²} . More precisely, for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> <m:mo>,</m:mo> <m:mi>C</m:mi> </m:mrow> <m:mo>⊂</m:mo> <m:msubsup> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> <m:mn>2</m:mn> </m:msubsup> </m:mrow> </m:math> {A,B,Cq²} , if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>A</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>B</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>C</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:msup> </m:mrow> <m:mo>≫</m:mo> <m:msup> <m:mi>q</m:mi> <m:mn>4</m:mn> </m:msup> </m:mrow> </m:math> {|A||B||C|1/2 q⁴} , then for any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>λ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msub> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> </m:msub> <m:mo>∖</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {λq\{0\}} , the number of congruence classes of triangles with vertices in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>A</m:mi> <m:mo>×</m:mo> <m:mi>B</m:mi> <m:mo>×</m:mo> <m:mi>C</m:mi> </m:mrow> </m:math> {A× B× C} and one side-length λ is at least <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi/> <m:mo>≫</m:mo> <m:msup> <m:mi>q</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> { q²} . In higher dimensions, we obtain similar results for k -simplex but under a slightly stronger condition. Compared to the well-known <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {L²} method in the literature, our approach offers better results in both conditions and conclusions. When <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>A</m:mi> <m:mo>=</m:mo> <m:mi>B</m:mi> <m:mo>=</m:mo> <m:mi>C</m:mi> </m:mrow> </m:math> {A=B=C} , the second goal of this paper is to give a new and unified proof of the best current results on the distribution of simplex due to Bennett, Hart, Iosevich, Pakianathan and Rudnev (2017) and McDonald (2020). The third goal of this paper is to study a Furstenberg-type problem associated to a set of rigid motions. The main ingredients in our proofs are incidence bounds between points and rigid motions. While the incidence bounds for large sets are due to the author and Semin Yoo (2023), the bound for small sets will be proved by using a point–line incidence bound in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> <m:mn>3</m:mn> </m:msubsup> </m:math> {Fq³} due to Kollár (2015).