Numbers with at most $2$ prime factors in short arithmetic progressions

TL;DR

This paper proves that under certain growth conditions, almost all short intervals contain numbers with at most two prime factors in specific arithmetic progressions.

Contribution

It establishes a new result on the distribution of almost primes with at most two prime factors in short arithmetic progressions under growth conditions.

Findings

Almost all short intervals contain numbers with ≤2 prime factors in given progressions.

The result holds when L/((q)  X  log X) o  as X   .

The theorem applies to a wide range of moduli and interval lengths.

Abstract

We show that if $\frac{L}{\varphi(q)\log X}\to\infty$ as $X\to\infty$, almost all $(a, x)\in (\mathbb Z/q\mathbb Z)^\times\times [X, 2X]$ are such that there exists a product of at most two primes in $[x, x + L]$ congruent to $a\mod{q}$.