Sumsets of the distance set in \mathbbF_q^d

2016/11/19 by Thang Pham, Pham, Thang
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1611.06398

openalex publication_date 2016/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let \mathbbF_q be a finite field of order q, where q is large odd prime power. In this paper, we improve some recent results on the additive energy of the distance set, and on sumsets of the distance set due to Shparlinski (2016). More precisely, we prove that for E⊆ \mathbbF_q^d, if d=2 and q^1+(1)/(4k-1)=o(|E|) then we have |kΔ_\mathbbF_q(E)|=(1-o(1))q; if d≥ 3 and q^{(d)/(2)+(1)/(2k)}=o(|E|) then we have |kΔ_\mathbbF_q(E)|=(1-o(1))q, where kΔ_\mathbbF_q(E):=Δ_\mathbbF_q(E)+⋯+Δ_\mathbbF_q(E) ~(k times).

Citations

• Some new inequalities in additive combinatorics
• Energies and structure of additive sets

Related

• Structural theorems on the distance sets over finite fields
• On the structure of distance sets over prime fields
• An asymmetric bound for sum of distance sets
• Erdos distance problem in vector spaces over finite fields
• The quotient set of the quadratic distance set over finite fields