New small gaps between squarefree numbers

TL;DR

This paper proves that small intervals of size proportional to x^{5/26} always contain a squarefree number for large x, improving understanding of the distribution of squarefree numbers.

Contribution

It establishes new asymptotic relations between shifts in quadratic forms and generalizes differencing techniques to analyze squarefree numbers.

Findings

Intervals of size C x^{5/26} contain squarefree numbers for large x

New asymptotic relations between shifts in quadratic forms are derived

Generalized differencing methods to study the distribution of squarefree numbers

Abstract

In this paper, we show that, for some constant $C > 0$, the interval $(x, x + C x^{5/26}]$ always contains a squarefree number when $x$ is sufficiently large (in terms of $C$). Our improvement comes from establishing asymptotic relations between the shifts $a$ and $b$ when $m n^2 \approx (m - a) (n + b)^2$ We apply them to study quadruples $(m + a_1) (n - b_1)^2 \approx m n^2 \approx (m - a_2)(n + b_2)^2 \approx (m - a_2 - a_3)(n + b_2 + b_3)^2$ and generalize Roth differencing and Filaseta-Trifonov differencing by allowing $b_1$ to be different from $b_3$. We also introduce a new differencing and exploit the interplay among these three differencings.