The k-resultant modulus set problem on algebraic varieties over finite fields

We study the k-resultant modulus set problem in the d-dimensional vector space Fqd over the finite field Fq with q elements. Given E⊂Fqd and an integer k≥2, the k-resultant modulus set, denoted by Δk(E), is defined asΔk(E)={‖x1±x2±⋯±xk‖∈Fq:xj∈E,j=1,2,…,k}, where ‖α‖=α12+⋯+αd2 for α=(α1,…,αd)∈Fqd. In this setting, the k-resultant modulus set problem is to determine the minimal cardinality of E⊂Fqd such that Δk(E)=Fq or Fq⁎. This problem is an extension of the Erdős-Falconer distance problem. In particular, we investigate the k-resultant modulus set problem with the restriction that the set E⊂Fqd is contained in a specific algebraic variety. Energy estimates play a crucial role in our proof.