Simultaneous diophantine approximation and asymptotic formulae on manifolds
Let \(\psi(q) \in \mathbb{N}\) for \(q=1, 2, 3, \dots\) be decreasing such that for some \(k \in \mathbb{N}\) the series \(\sum\psi(q)^k\) is divergent. As a slight variation of Khintchine's theorem on simultaneous diophantine approximation, it is known that, for almost all \((x_1, \dots, x_k) \in\mathbb{R}^k\) there are infinitely many solutions of the system of inequalities \[ |qx_i |< \psi(q), \quad i=1, \dots, k \tag{*} \] where \(|r|= \min \{|r-m|: m\in \mathbb{Z}\}\). In this paper the authors continue their studies in a series of papers from the case for convergent \(\sum \psi (q)^k\) to the above divergent case [see, for example, Acta Arith. 57, 115-130 (1991; Zbl 0736.11040)]. They apply a powerful variance argument due to \textit{W. M. Schmidt} [Trans. Am. Math. Soc. 110, 493-518 (1964; Zbl 0199.09402)] to obtain an asymptotic formula for the number of integer solutions \(q\) of the system (*) for almost all points \((x_1, \dots, x_k)\) on a smooth submanifold \(M\) of \(\mathbb{R}^k\) which satisfies certain curvature conditions entailing restrictions on the codimension of \(M\).
number of integer solutions on a smooth submanifold
scientific article; zbMATH DE number 894609