Smooth billiards with a large Weyl remainder

TL;DR

This paper establishes lower bounds for the error term in Weyl's law for smooth, star-shaped planar domains with periodic billiard trajectories, extending classical results and confirming Sarnak's predictions.

Contribution

It proves new lower bounds for the Weyl remainder in smooth domains with periodic billiard trajectories, including higher-dimensional cases, generalizing previous boundaryless surface results.

Findings

Lower bounds for Weyl remainder in smooth star-shaped domains.

Examples include ellipses and constant width domains.

Results extend to higher dimensions with specific domain conditions.

Abstract

The celebrated Hardy-Landau lower bound for the error term in the Gauss's circle problem can be viewed as an estimate from below for the remainder in Weyl's law on a square, with either Dirichlet or Neumann boundary conditions. We prove an analogous estimate for smooth star-shaped planar domains admitting an appropriate one-parameter family of periodic billiard trajectories. Examples include ellipses and smooth domains of constant width. Our results confirm a prediction of P. Sarnak who proved a similar statement for surfaces without boundary. We also obtain lower bounds on the error term in higher dimensions. In this case, the main contribution to the Weyl remainder typically comes from the "big" singularity at zero of the wave trace. However, for certain domains, such as the Euclidean ball, the dimension of the family of periodic trajectories is large enough to dominate the contribution of the singularity at zero.