Existence of Smooth Numbers in Short Intervals

ABSTRACT Let $X \ge y \ge 2$, and let $u = \frac{\log X}{\log y}$. We say a number is 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 $y \ge \exp \left( (\log X)^{2/3 + \varepsilon } \right)$, we show that the interval $[x, x+h]$ contains a y-smooth number for almost all $x \in [X, 2X]$, provided $h \ge \exp \left( (1 + \varepsilon ) \left( \frac{11}{8} u \log u + 4 \log \log X \right) \right)$, and $X$ is sufficiently large depending on $\varepsilon$. This result improves upon an earlier result by Matomäki. Additionally, we provide the corresponding ‘all intervals’ type result. Our approach relies on a strategically factorized Dirichlet polynomial, much like the earlier work of Matomäki. The improvement in our results stems from the integration of ideas introduced in the breakthrough work of Matomäki and Radziwiłł.