Consider billiard dynamics in a strictly convex domain, and consider a trajectory that begins with the velocity vector making a small positive angle with the boundary. Lazutkin proved that in two dimensions, it is impossible for this angle to tend to zero along trajectories. We prove that such trajectories can exist in higher dimensions. Namely, using the geometric techniques of Arnold diffusion, we show that in three or more dimensions, assuming the geodesic flow on the boundary of the domain has a hyperbolic periodic orbit and a transverse homoclinic, the existence of trajectories asymptotically approaching the billiard boundary is a generic phenomenon in the real-analytic topology.

We prove a Lindelöf-on-average upper bound for the fourth moment of Dirichlet L-functions of conductor q along a coset of the subgroup of characters modulo d when $q^{\ast }\mid d$, where $q^{\ast }$ is the least positive integer such that $q^2\mid {(q^{\ast })}^3$. As a consequence, we finish the previous work of the authors and establish a Weyl-strength subconvex bound for all Dirichlet L-functions with no restrictions on the conductor.

The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter Δ. When $\mathrm{\Delta }=0$, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all Δ, there has been very little progress in understanding its statistics in the scaling limit for other values. In this work, we focus on the six-vertex model with domain wall boundary conditions at $\mathrm{\Delta }=1∕2$, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We show that the maximum of the topmost level line for a uniformly random ASMs has the Gaussian orthogonal ensemble (GOE) Tracy–Widom distribution after appropriate rescaling. A key ingredient in our proof is Zeilberger’s proof of the ASM conjecture. As far as we know, this is the first edge fluctuation result away from the tangency points for the domain-wall six-vertex model when we are not in the free-fermion case.