On the difference between consecutive squarefree integers

The Distribution of Fractional Parts with Applications to Gap Results in Number Theory

“…Theorem 1 for k = 2 was first established by the authors in [11] and for general k was first established by the second author in [49]. For k = 2, this result improves on work by Fogels [13], Roth [39], Richert [38], Rankin [37], Schmidt [41], Graham and Kolesnik [16], the second author [46,47], the first author [6], and the authors jointly [10]. For general k, this result improves on work by Halberstam and Roth [18], the first author [5], and the second author [48].…”

“…This equation holds for every choice of / and /' with O^y < / ' «£ 2k. With / = 0 and ;' = 1, the equation is analogous to the situation we had in (16). A precisely analogous argument to what followed in the remainder of that paragraph leads us to deduce that we cannot have the 2/c + 1 numbers p, p + a u ..., p + a, + ... + a 2k above; in other words, we obtain a contradiction to the assumption that (i) and (ii) do not hold.…”

Klaus Friedrich Roth, 1925-2015

“…There followed papers by Richert 〈〉, Rankin 〈〉, Schmidt 〈〉, Graham and Kolesnik 〈〉, Trifonov 〈〉 and Filaseta 〈〉, many of them quite technical.…”

ON THE DIVISORS OF ORDER r

“…Estimates for b 16 are given in Table 1. Estimate for b = 1 belongs to Huxley [5], and estimate for b = 2 belongs to Graham and Kolesnik [4]. We have found no references on the best known results for b 3, so we calculated them with the use of [8, Th.…”