Cusp Excursions, Lattice Points on Manifolds, and the Mizohata-Takeuchi Conjecture

Abstract

We prove new logarithm laws for cusp excursions in spaces of lattices, and produce quantitative lower bounds for lattice points near submanifolds, using tools from dynamics and the geometry of numbers. As an application, we provide a new proof of power loss for the local Mizohata-Takeuchi conjecture with explicit error terms, as well as show that power loss is generic in $C^k$. The construction uses high-dimensional probabilistic estimates, but replaces the random orthogonal subspaces of Cairo-Zhang with random unimodular lattices; this yields stronger bounds and provides a richer family of counterexamples.