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.