Somewhat smooth numbers in short intervals

TL;DR

This paper demonstrates the abundance of smooth numbers in short intervals using exponent pairs, and applies the results to show many practical numbers appear in such intervals under certain conditions.

Contribution

It introduces new bounds for the existence of smooth numbers in short intervals using exponent pairs, including an unconditional result and a conjectural improvement.

Findings

Many $x^a$-smooth numbers exist in short intervals $[x-x^b,x]$ for $a>1/2$

Unconditional admissible $b$ is given by $b=1-a-a(1-a)^3$

Under the exponent-pairs conjecture, $b$ can be improved to $(1-a)/2+ ext{epsilon}

Abstract

We use exponent pairs to establish the existence of many $x^a$-smooth numbers in short intervals $[x-x^b,x]$, when $a>1/2$. In particular, $b=1-a-a(1-a)^3$ is admissible. Assuming the exponent-pairs conjecture, one can take $b=(1-a)/2+\epsilon$. As an application, we show that $[x-x^{0.4872},x]$ contains many practical numbers when $x$ is large.