Distinct distances on regular varieties over finite fields

In this paper we study some generalized versions of a recent result due to Covert, Koh, and Pi (2015). More precisely, we prove that if a subset E in a regular variety satisfies |E|≫qd−12+1k−1, thenΔk,F(E):={F(x1+⋯+xk):xi∈E,1≤i≤k}⊇Fq∖{0}, for some certain families of polynomials F(x)∈Fq[x1,…,xd].