A Delsarte Linear Programming Approach to the Erd\H{o}s--Falconer Distance Problem over Finite Fields

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
\[
\Delta_Q(E)=\{Q(x-y):\ x,y\in E\}.
\]
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.