  An interesting result of Veech more than 50 years ago is a parity, or mod
$2$, version of the Kronecker--Weyl equidistribution theorem concerning the
irrational rotation sequence $\{q\alpha\}$, $q=0,1,2,3,\ldots.$ If $\alpha$ is
badly approximable and $b\in(0,1)$ satisfies $b\ne\{m\alpha\}$ for any
$m\in\mathbb{Z}$, then the parity of cardinalities of the sets $\{1\le q\le
N:\{q\alpha\}\in[0,b)\}$ as $N\to\infty$ is evenly distributed.
  We first answer a question of Veech and establish a stronger form of the mod
$n$ analog of his result (Theorem 3.1). Furthermore, for irrational $\alpha$
and $b=\{m\alpha\}$ for some $m\in\mathbb{N}$, we give a simple yet precise
characterization of those cases that give rise to even distribution (Theorem
2.1). We also obtain time-quantitative description of some very striking
violations of uniformity -- this part is particularly number theoretic in
nature, and involves Ostrowski representations of positive integers and
$\alpha$-expansions of real numbers (Theorem 3.4).
  The Veech discrete $2$-circle problem can also be visualized as a problem
that concerns $1$-direction geodesic flow on a surface obtained by modifying
the surface comprising two side-by-side squares by the inclusion of symmetric
barriers and gates on the vertical edges, with appropriate modification of the
vertical edge identifications. We establish a far-reaching generalization of
this case to ones that concern $1$-direction geodesic flow on surfaces obtained
by modifying a finite square tiled translation surface in analogous but not
necessarily symmetric ways (Theorem 3.2).
