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.

arXiv commentThere is an error in the article. The recursive argument for Proposition 4 does not work if there is only one element in a "short interval"