Simultaneous diophantine approximation of rationals by rationals (Q1083475)

For positive integers \(n\geq 2\), \(B\geq 2\), let \(S_ n(B)\) denote the set of rational vectors \(\alpha =(a_ 1/B,...,a_ n/B)\) with \(a_ j\in {\mathbb{Z}}\), \(0\leq a_ j<B\), \(\gcd (a_ 1,...,a_ n,B)=1.\) Furthermore, for \(\Delta >0\), define N(\(\alpha\),\(\Delta)\) as the number of vectors \(\zeta =(x_ 1/x,...,x_ n/x)\) with \(1\leq x<B\), \(x_ j\) and x integers, which are good approximations for \(\alpha\), in the sense that \(| a_ i/B-x_ i/x| \leq \Delta /Bx\) for \(i=1,...,n\). This paper provides estimates for the first and second moments of N(\(\alpha\),\(\Delta)\) over \(S_ n(B)\). The main results read \[ \sum_{\alpha \in S_ n(B)}N(\alpha,\Delta)=(2\Delta)^ n (B- 1)\prod_{p| B}(1-p^{-n})+\theta 2n 3^{n-1} d(B)B \Delta^{n- 1}\quad (| \theta | <1) \] where \(d(B)=\sum_{d| B}1\), and \[ \sum_{\alpha \in S_ n(B)}N(\alpha,\Delta)^ 2 \ll B^{2-n} \Delta^{2n}+B \Delta^ n+\lambda_ n d(B)B \Delta^ n \] where \(\lambda_ 2=\log \Delta\), \(\lambda_ 3=\lambda_ 4=1\), \(\lambda_ n=0\) for \(n\geq 5\). This second estimate is not far away from best possible, since it is shown that, for \(B=p\) a prime, \[ \sum_{\alpha \in S_ n(p)}N(\alpha,\Delta)^ 2 \gg p^{2-n} \Delta^{2n}+p \Delta^ n. \]