A Groshev type theorem for convergence on manifolds
Diophantine approximations on so-called non-degenerate manifolds are considered. It is proved: Theorem: Let \(n\geq 2\), \(m\in \mathbb N\) or \(m=\infty\), \(d\in \mathbb N\) with \(d\leq n\), \(U\subset \mathbb R^d\) be an open set and \(g:U\to \mathbb R^n\) be a \(C^{(m)}\) non-degenerate map. For any monotonic \(\psi:\mathbb N\to \mathbb R^+\) such that the sum \[ \sum^\infty_{h=1} h^{n-1}\psi(h) \] converges, the inequality \[ \|a\cdot g(x)\|=\|a_n g_n(x)+\dots +a_1 g_1(x)\|<\psi(|a|_\infty) \] has at most finitely many solutions \(a\in \mathbb Z^n\) for almost all \(x\in U\). \smallskip Remarks: \begin{itemize} \item[1.] \(\|x\|=\inf\{|x-p|:p\in\mathbb Z\}\) \ for \(x\in \mathbb R\) \(a\cdot b=\sum\limits^n_{i=1}a_i\cdot b_i,\;|a|=\sqrt{a\cdot a}\) \(|a|_\infty=\max\{|a_i|:i=1,\dots,n\}\), where \(a=(a_1,\dots,a_n),b=(b_1,\dots,b_n) \in \mathbb R^n\). \item [2.] The proof uses a new approach which further develops a classical argument of Sprindzhuk. \item [3.] The paper contains a good introduction with many citations on former papers concerning results of Khintchine and Groshev type theorems.\end{itemize}
scientific article; zbMATH DE number 1822364