Erd\H{o}s--Falconer distance conjecture from an analytic perspective

Let \(q\) be an odd prime power and let \(V=\F_q^{2m}\), equipped with \(Q(x)=x_1^2+\cdots+x_{2m}^2\). We develop a semidefinite Delsarte framework for the two-set Erdős--Falconer distance problem over \(V\). The framework reduces the natural \(q^m\)-scale positive-proportion theorem to a uniform \(L^1\) anti-concentration statement for positive convex combinations of classical Kloosterman sums. Assuming this Kloosterman anti-concentration conjecture, we prove that for every \(0<\alpha<\frac{1}{2}\) there is a constant \(C_{m,\alpha}\) such that \[
\min\{|E|,|F| \} \ge C_{m,\alpha}q^m
\quad\Longrightarrow\quad
|\Delta_Q^\times(E,F)|>\alpha(q-1) \] for all \(E,F\subset \F_q^{2m}\). More generally, a \(q^{-\theta}\)-level version of the Kloosterman input yields the geometric threshold \(q^{m+\theta}\). In particular, a universal second-moment argument gives an unconditional \(q^{m+\frac{1}{2}}\)-threshold through the same framework. The proof uses positive semidefinite \(2\times2\) Gram matrices on quadratic frequency shells, the shell Fourier transform in even dimension, and a minimax separation argument that produces a uniform signed combination of Kloosterman columns. We also provide evidence for the Kloosterman conjecture and discuss limitations near full support.