A Delsarte Linear Programming Approach to the Erdős--Falconer Distance Problem over Finite Fields

Abstract

We introduce a Delsarte linear programming approach to the finite field Erdős--Falconer distance problem. Let \(q\) be an odd prime power, let \(n\) be even, and let \(Q\) be a non-degenerate quadratic form on \(\mathbb{F}_q^n\). For \(E\subset \mathbb{F}_q^n\), define \[ Δ_Q(E)=\{Q(x-y):\ x,y\in E\}. \] We prove that, for every fixed \(0<α<\frac{1}{2}\), there exist constants \(C_α>0\) and \(q_α\) such that if \(q\ge q_α\) and $|E|\ge C_αq^{\frac n2+\frac13},$ then \[ |Δ_Q(E)|>1+α(q-1). \] In particular, \(Δ_Q(E)\) contains a positive proportion of the elements of \(\mathbb{F}_q\), and hence \(|Δ_Q(E)|\gg q\). Our result applies uniformly to all non-degenerate quadratic forms in even-dimensional finite field vector spaces. In the Euclidean case \[ Q(x)=x_1^2+\cdots+x_n^2, \] it improves, for every even \(n\ge 4\) over arbitrary finite fields, the general exponent \(\frac{n+1}{2}\) obtained by Iosevich and Rudnev to $\frac n2+\frac13.$ The proof is based on the association scheme arising from the level sets of \(Q\). By analyzing the corresponding eigenvalues through Gauss sums and Kloosterman sums, we construct a suitable feasible solution to the Delsarte linear program. This provides a new algebraic-combinatorial method for obtaining distance set estimates over finite fields.