Smooth Numbers in Short Intervals

Let $X\ge y\ge 2$, and let $u=\frac{\log X}{\log y}$. We say a number is $y\mathit{\text{-smooth}}$ 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 $y\ge \exp ((\log X{)}^{2\mathrm{/}3+\epsilon })$, we show that the interval $[x,x+h]$ contains a $y$-smooth number for almost all $x\in [X,2X]$, provided $h\ge \exp ((1+\epsilon )(\frac{11}{8}u\log u+4\log \log X))$, and $X$ is sufficiently large depending on $\epsilon$. This result improves upon an earlier result by Matomäki. Additionally, we provide the corresponding ``all intervals" type result.