  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.
