On the sums of any k points in finite fields

2014/03/24 by David Covert, Covert, David, Doowon Koh +3
Computer Science · Mathematics · #11T23 #42B05 #52C10 #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 #Number Theory (math.NT) #math.CA #math.CO #math.NT #msc:11T23 #msc:42B05 #msc:52C10

paper · pdf · doi:10.48550/arxiv.1403.6138

16 pages

openalex publication_date 2014/03/24 · arxiv created 2015/02/04 · arxiv updated 2015/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a set E⊂ \mathbb F_q^d, we define the k-resultant magnitude set as Δ_k(E) =\‖x₁ + … + x_k‖∈ \mathbb F_q: x₁, …, x_k ∈ E\, where ‖v‖=v₁²+⋯+ v_d² for v=(v₁, …, v_d) ∈ \mathbb F_q^d. In this paper we find a connection between a lower bound of the cardinality of the k-resultant magnitude set and the restriction theorem for spheres in finite fields. As a consequence, it is shown that if E⊂ \mathbb F_q^d with |E|≥ C q^{(d+1)/(2)-(1)/(6d+2)}, then |Δ₃(E)|≥ c q for d = 4 or d = 6, and |Δ₄(E)| ≥ cq for even dimensions d ≥ 8. In addition, we prove that if d≥ 8 is even, and |E|≥ C_ε ~q^{(d+1)/(2) - (1)/(9d -18) + ε} for ε >0, then |Δ₃(E)|≥ c q.

Citations

• Extension theorems for the Fourier transform associated with non-degenerate quadratic surfaces in vector spaces over finite fields

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