Almost primes in almost all very short intervals

TL;DR

The paper proves that for large enough intervals, almost all contain a number with at most two prime factors, using advanced sieve methods and results on Kloosterman sums.

Contribution

It introduces a new result showing almost all very short intervals contain almost primes, employing Richert's weighted sieve and deep exponential sum estimates.

Findings

Almost all intervals of the form (x - h log X, x] contain an almost prime for large h and X.

The proof combines sieve techniques with bounds on Kloosterman sums.

The result extends understanding of the distribution of almost primes in short intervals.

Abstract

We show that as soon as $h\to \infty$ with $X \to \infty$, almost all intervals $(x-h\log X, x]$ with $x \in (X/2, X]$ contain a product of at most two primes. In the proof we use Richert's weighted sieve, with the arithmetic information eventually coming from results of Deshouillers and Iwaniec on averages of Kloosterman sums.