Khinchin's theorem in \(k\) dimensions with prime numerator and denominator (Q2759129)

This paper deals with a generalization of a theorem of Khintchine, stating that if \(\psi(x)\) is a positive function such that for increasing \(x\) \(x\psi(x)\) is decreasing, then, for almost all \(\alpha\) (that is, for all \(\alpha\) in \((0,1)\), except possibly a set of measure 0) the inequality NEWLINE\[NEWLINE\left|\alpha -\frac mn\right|< \frac{\psi(n)}{n}\tag{*}NEWLINE\]NEWLINE has an infinity of solutions with integer \(m\) and natural \(n\) if and only if NEWLINE\[NEWLINE\sum^\infty_1\psi(n)=\infty.\tag{**}NEWLINE\]NEWLINE The generalization goes in two directions. Firstly, for the solvability of (*) for almost all \(\alpha\) stronger restrictions are imposed, than \(m\) be integer and \(n\) be natural. It is required that \(m\) and \(n\) be prime numbers. As is expected, in this case (**) is replaced by a stronger restriction, namely by the divergence of the series \(\sum_n^\infty \frac{\psi(n)}{\log n}\). NEWLINENEWLINENEWLINEThe second generalization is that to several dimensions. The general theorem is as follows.NEWLINENEWLINENEWLINETheorem. Let \(\psi_1(n),\psi_2(n),\dots,\psi_k(n)\) be decreasing functions of \(n\), \(\underline{\alpha} = (\alpha_1,\alpha_2,\dots,\alpha_k)\) a point in the \(n\)-dimensional unit cube. Then the necessary and sufficient condition for the solvabilily of the simultaneous inequalities NEWLINE\[NEWLINE\left|\alpha_1 - \frac{q_i}{p}\right|< \frac{\psi_i(p)}{p}\quad (i=1,2,\dots,k; \;q_1,q_2,\dots,q_k, p\text{ primes})NEWLINE\]NEWLINE for almost all \(\underline{\alpha}\) with infinitely many \((n + 1)\)-tuples is the divergence of the series NEWLINE\[NEWLINE\sum^\infty_2 \frac{\psi_1(n)\psi_2(n)\dots \psi_\lambda(n)}{(\log n)^{k+1}} .NEWLINE\]NEWLINE The main idea of the proof is related to that of the proof of Khintchine for his proof of the corresponding multidimensional theorem, which he published some two years after the one-dimensional one and which avoids the theory of continued fractions. However, because of the primality of \(p\) and the \(q_i\)'s, results from the theory of primes are also needed.