2015/08/11 by David Covert, Covert, David, Doowon Koh +3
Computer Science · Mathematics · #42B05 #43A15 #43A32 #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #math.CA #math.CO #msc:42B05 #msc:43A15 #msc:43A32
paper · pdf · doi:10.48550/arxiv.1508.02688
arxiv created 2015/08/11 · openalex publication_date 2015/08/11 · arxiv updated 2015/08/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We study the k-resultant modulus set problem in the d-dimensional vector space \mathbb Fqd over the finite field \mathbb Fq with q elements. Given E⊂ \mathbb Fqd and an integer k≥ 2, the k-resultant modulus set, denoted by Δk(E), is defined as Δk(E)=\‖x1± x2 ± ⋯ ± xk‖∈ \mathbb Fq: xj∈ E, ~j=1,2,…, k\, where ‖α‖=α12+⋯+ αd2 for α=(α1, …, αd) ∈ \mathbb Fqd. In this setting, the k-resultant modulus set problem is to determine the minimal cardinality of E⊂ \mathbb Fqd such that Δk(E) = \mathbb Fq or \mathbbFq^*. 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⊂ \mathbb Fqd is contained in a specific algebraic variety. Energy estimates play a crucial role in our proof.