  We consider the distribution of the binomial probability mass function (pmf)
among arithmetic progressions and obtain an average-type theorem. As
applications, we consider the possible visits to a kind of sieved sets of
integers or lattice points, by an $\alpha$-random walker. We show that, almost
surely, the asymptotic proportion of time the random walker in a sieved set of
the type is equal to the density of the set, independently of $\alpha$.

Number Theory