New small gaps between 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.