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

We introduce a Delsarte linear programming approach to the finite field Erd\H{o}s--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 $$\Delta_Q(E)=\{Q(x-y):\ x,y\in E\}.$$ We prove that, for every fixed $0<\alpha<\frac{1}{2}$, there exist constants $C_\alpha>0$ and $q_\alpha$ such that if $q\ge q_\alpha$ and $|E|\ge C_\alpha q^{\frac n2+\frac13},$ then $$|\Delta_Q(E)|>1+\alpha(q-1).$$ In particular, $\Delta_Q(E)$ contains a positive proportion of the elements of $\mathbb{F}_q$, and hence $|\Delta_Q(E)|\gg q$. 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.