Smooth Numbers in Short Intervals

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.