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 | ≫ q d − 1 2 + 1 k − 1 , then Δ k , F ( E ) : = { F ( x 1 + ⋯ + x k ) : x i ∈ E , 1 ≤ i ≤ k } ⊇ F q ∖ { 0 } , for some certain families of polynomials F ( x ) ∈ F q [ x 1 , … , x d ] . [ABSTRACT FROM AUTHOR]