Erdős--Falconer distance conjecture from an analytic perspective

Let $q$ be an odd prime power and let $V={\text{F}}_q^{2m}$, 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 exists a constant $C_{m,\alpha }$ such that $$\min \{\mathrm{\mid }E\mathrm{\mid },\mathrm{\mid }F\mathrm{\mid }\}\ge C_{m,\alpha }{q}^m⟹\mathrm{\mid }{\mathrm{\Delta }}_Q^{\times }(E,F)\mathrm{\mid }>\alpha (q-1)$$ for all $E,F\subset {\text{F}}_q^{2m}$. More generally, a $q^{-\theta }$-level version of the Kloosterman input yields the geometric threshold $q^{m+\theta }$. Unconditionally, we establish a uniform $q^{-\frac{1}{3}}$-level $L^1$ lower bound whenever the support has size at most $\alpha (q-1)$, with $0<\alpha <\frac{1}{2}$. Consequently, $$\min \{\mathrm{\mid }E\mathrm{\mid },\mathrm{\mid }F\mathrm{\mid }\}\ge C_{m,\alpha }{q}^{m+\frac{1}{3}}⟹\mathrm{\mid }{\mathrm{\Delta }}_Q^{\times }(E,F)\mathrm{\mid }>\alpha (q-1)\mathrm{.}$$ The proof uses positive semidefinite $2\times 2$ 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 constant-scale Kloosterman conjecture and discuss its limitations near full support.