Polynomial patterns in subsets of large finite fields of low characteristic

2023/03/02 by Ackelsberg, Ethan, Bergelson, Vitaly
#11B30 (Primary) 11T06 #37A25 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2303.00925

Abstract

We prove a low characteristic counterpart to the main result in (Peluse, 2019), establishing power saving bounds for the polynomial Szemerédi theorem for certain families of polynomials. Namely, we show that if P₁, …, P_m ∈ (\mathbbF_p[t])[y] satisfy an equidistribution condition, which is a natural variant of the independence condition in (Peluse, 2019) for our context, then there exists γ> 0 such that for any q = p^k and any A₀, A₁, …, A_m ⊆ \mathbbF_q, | \ (x,y) ∈ \mathbbF_q² : x ∈ A₀, x + P₁(y) ∈ A₁, …, x + P_m(y) ∈ A_m \ | = q^-(m-1) ∏_i=0^m|A_i| + O_{q → ∞; P1, …, Pm} ( |A₀|^1/2 q^{3/2 - γ} ). In particular, if A ⊆ \mathbbF_q contains no pattern \x, x + P₁(y), …, x + P_m(y)\ of cardinality m+1, then |A| ≪_{P1, …, Pm} q^{1 - γ/ ( m + (1)/(2) )}.

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