On the proximity of divisors of integers
The authors consider the interesting problem of finding conditions on \(\lambda\geq 0\) such that, for almost all \(r\)-tuples \((n_1,\dots, n_r)\) of positive integers, there exist divisors \(d_j|n_j\) \((1\leq r)\) with \[ 0<\log (d_j/d_1)\leq (\log n_1)^{-\lambda}, \qquad 2\leq j\leq r. \] \textit{A. Raouj} considered the case \(r=2\) in [Acta Arith. 69, 121-152, 171-188 (1995; Zbl 0821.11045)] where it is shown that the statement above holds for \(\lambda< \lambda_2^*= \log 4-1\) and fails for \(\lambda> \lambda_2^*\). In this paper the authors tackle the cases \(r=3\), \(r\geq 4\). They prove that when \(r=3\) the critical value of \(\lambda\) is \(\lambda_3^*= -1+(3/2)\log 2\), in the sense that the statement above holds for \(\lambda< \lambda_3^*\) and fails for \(\lambda> \lambda_3^*\). However when \(r\geq 4\) they show that the statement fails for all \(\lambda\geq 0\). In all cases an estimate of the number of \(r\)-tuples \((n_1,\dots, n_r)\) with \(n_j\leq x_j\) that lie in the exceptional set is obtained. Inevitably the proof is very technical and depends on a number of subsidiary results.
scientific article; zbMATH DE number 1310085