Conditional expanding bounds for two-variable functions over finite\n valuation rings

2015/10/12 by Le Quang Ham, Ham, Le Quang, Thang V. Pham +3
Mathematics · #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1510.03479

openalex publication_date 2015/10/12 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

In this paper, we use methods from spectral graph theory to obtain some\nresults on the sum-product problem over finite valuation rings \R of\norder q^r which generalize recent results given by Hegyv 'ari and Hennecart\n(2013). More precisely, we prove that, for related pairs of two-variable\nfunctions f(x,y) and g(x,y), if A and B are two sets in \R^*\nwith |A|=|B|=q^\α, then \
max
left
lbrace |f(A, B)|, |g(A, B)|\n
right
rbrace
gtrsim |A|^{1+
Delta(
alpha)}, for some \Δ(\α)>0.\n

Related

• On three-variable expanders over finite valuation rings
• A note on sum-product estimates over finite valuation rings
• Sum-Product Type Estimates for Subsets of Finite Valuation Rings
• Algebraic methods in sum-product phenomena
• On the maximum values of the additive representation functions