Smooth Numbers in Short Intervals

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.