Matthew Welsh

Pair correlation for roots of quadratic congruences

For a fixed integer $D$, we consider the set of $\frac{\mu}{m} \pmod 1$ where $\mu \pmod m$ satisfies $\mu^2 \equiv D \pmod m$ and $m \leq M$. Hooley (1963) showed that in the limit as $M \to \infty$ this set becomes equidistributed modulo 1 using input from algebraic geometry, and Bykovskii (1983) refined this result using the spectral theory of automorphic forms. In joint work with Jens Marklof, we use ergodic theoretic techniques to show the existence of limits for a large class of fine-scale statistics (including a new proof of equidistribution) and find an explicit expression for the pair correlation.