On Gaps Between Squarefree Numbers II

In this paper, the authors continue their work on the problem of nding an h = h(x) as small as possible such that for x su ciently large, there is a squarefree number in the interval (x; x + h]: This problem has been investigated by Fogels [4], Roth [11], Richert [10], Rankin [9], Schmidt [12], Graham and Kolesnik [5], the second author [14,15], and the rst author [2]. In particular, the authors [3] have recently shown by elementary means that there is a constant c > 0 such that for x su ciently large, the interval (x; x+ h] with h = cx contains a squarefree number. Using exponential sums, they showed that 8/37 may be replaced by 3/14. A more extensive history of the problem can be found in their paper [3]. The purpose of this paper is to make the following improvement.