In this paper we prove a sharp condition to guarantee of having a positive
proportion of all congruence classes of triangles in given sets in
Fq2β. More precisely, for A,B,CβFq2β, if
β£Aβ£β£Bβ£β£Cβ£1/2β«q4, then for any Ξ»βFqββ{0}, the number of congruence classes of triangles with vertices in AΓBΓC and one side-length Ξ» is at least β«q2. As a
consequence, the number of congruence classes of triangles with vertices in
AΓBΓC is at least β«q3. The main ingredients in our proof
are a recent incidence bound between points and rigid motions due to the author
and Semin Yoo (2023) and a result on a Furstenberg type problem. When three
sets are the same, we give a unified and new proof for all the best current
results due to Bennett, Hart, Iosevich, Pakianathan, and Rudnev (2017) and
McDonald (2020). The novelty of this approach is to present an application of
results on the number of k-rich rigid motions in studying the distribution of
simplex. A number of related questions will be also addressed in this paper.Comment: 21 page