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

Abstract

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 \(0<\alpha<\frac12\), there exists 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 \(q^{m+\theta}\). Unconditionally, we establish a uniform \(q^{-\frac13}\)-level \(L^1\) lower bound whenever the support has size at most \(\alpha(q-1)\), with \(0<\alpha<\frac12\). Consequently, \[ \min\{|E|,|F|\}\ge C_{m,\alpha}q^{m+\frac13} \quad\Longrightarrow\quad |\Delta_Q^\times(E,F)|>\alpha(q-1). \] 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 constant-scale Kloosterman conjecture and discuss its limitations near full support.