Binomial pmf among arithmetic progressions and sieved sets in random walks

TL;DR

This paper studies the distribution of binomial pmf in arithmetic progressions and applies it to analyze the visits of an $ ext{alpha}$-random walker to sieved sets, showing the proportion of visits matches the set density almost surely.

Contribution

It introduces an average-type theorem for binomial pmf among arithmetic progressions and establishes almost sure visitation proportions for sieved sets by random walks.

Findings

The asymptotic visitation proportion equals the set density.

Almost sure results hold for the random walk visits.

The approach links binomial distribution properties with random walk behavior.

Abstract

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 $\alpha$.