  We study the statistical properties of the counting function of lattice
points inside thin annuli. By a conjecture of Bleher and Lebowitz, if the width
shrinks to zero, but the area converges to infinity, the distribution converges
to the Gaussian distribution. If the width shrinks slowly to zero, the
conjecture was proven by Hughes and Rudnick for the standard lattice, and in
our previous paper for generic rectangular lattices. We prove this conjecture
for arbitrary lattices satisfying some generic Diophantine properties, again
assuming the width of the annuli shrinks slowly to zero.
  One of the obstacles of applying the technique of Hughes-Rudnick on this
problem is the existence of so-called close pairs of lattice points. In order
to overcome this difficulty, we bound the rate of occurence of this phenomenon
by extending some of the work of Eskin-Margulis-Mozes on the quantitative
Openheim conjecture.
