Smooth Numbers in Short Intervals

TL;DR

This paper investigates the distribution of y-smooth numbers within short intervals, establishing new bounds that ensure their presence in almost all such intervals for large X, improving previous results by Matomäki.

Contribution

The paper provides improved bounds on the length of short intervals containing y-smooth numbers, extending the understanding of their distribution in almost all intervals for large X.

Findings

Almost all intervals of specified length contain a y-smooth number.

Improved bounds over previous results by Matomäki.

Applicable for y ≥ exp((log X)^{2/3 + ε}) and large X.

Abstract

Let \( X \geq y \geq 2 \), and let \( u = \frac{\log X}{\log y} \). We say a number is \textit{$y$-smooth} if all of its prime factors are less than or equal to \( y \). In this paper, we study the distribution of $y$-smooth numbers in short intervals. In particular, for \( y \geq \exp\left( (\log X)^{2/3 + \epsilon} \right) \), we show that the interval \( [x, x+h] \) contains a $y$-smooth number for almost all \( x \in [X, 2X] \), provided \( h \geq \exp\left( (1 + \epsilon) \left( \frac{11}{8} u \log u + 4 \log \log X \right) \right) \), and \( X \) is sufficiently large depending on \( \epsilon \). This result improves upon an earlier result by Matom\"aki. Additionally, we provide the corresponding ``all intervals" type result.