Khintchine-type theorems on manifolds: The convergence case for standard and multiplicative versions (Q2726332)

In this important paper, an analogue of the Khintchine-Groshev theorem in the case of convergence is proved for nondegenerate smooth manifolds in \(\mathbb{R}^n\). NEWLINENEWLINENEWLINEMore precisely, let \(U\subset\mathbb{R}^d\) be open and let \(f:U\to \mathbb{R}^n\) be a nondegenerate \(n\)-tuple of \(C^m\) functions. Write \(|\langle x\rangle|= \min\{|x-k|: k\in \mathbb{Z}\}\) and \(\|{\mathbf y}\|= \max\{|y_1|,\dots, |y_n|\}\). It is shown using the geometry of lattices in Euclidean spaces and methods from metric Diophantine approximation that if the function \(\Psi:\mathbb{Z}^n\setminus \{0\}\to (0,\infty)\) satisfies a coordinatewise nonincreasing condition and if the sum \(\sum_{{\mathbf q}\in \mathbb{Z}^n\setminus \{0\}} \Psi({\mathbf q})\) converges, then the set NEWLINE\[NEWLINE{\mathcal W}(\Psi)= \{{\mathbf x}\in U: |\langle{\mathbf q}.f({\mathbf x})\rangle|\leq \Psi({\mathbf q}) \text{ for infinitely many }{\mathbf q}\in \mathbb{Z}\}NEWLINE\]NEWLINE has Lebesgue measure 0. The standard case, in which \(\Psi({\mathbf q})= \psi(\|{\mathbf q}\|^n)\), and the multiplicative case, in which \(\psi({\mathbf q})= \psi(\prod_{j=1}^n (\max \{1,|q_j|\}))\), hold if \(\psi(q)\) is nonincreasing and \(\sum_{q\in \mathbb{N}}\psi(q)< \infty\) and \(\sum_{q\in \mathbb{N}}(\log q)^{n-1} \psi(q)< \infty\), respectively. NEWLINENEWLINENEWLINEThe case where \(\psi(x)= x^{-1-\varepsilon}\), \(\varepsilon>0\), was conjectured by Sprindzhuk for analytic manifolds and was proved by the second and third authors for smooth manifolds, using ideas from dynamical systems [Ann. Math. (2) 148, 339-360 (1998; Zbl 0922.11061)]. \textit{V. V. Beresnevich} has used Sprindzhuk's method of `essential and inessential' domains to obtain the convergence result for the standard case [A Groshev-type theorem for convergence on manifolds, to appear in Acta Math. Hung.]. Recently, the even more difficult case of divergence has been established jointly by Beresnevich and the three authors.