Erdős--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 Erd\H{o}s--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.