Metric theory of Diophantine approximation in the field of complex numbers (Q2707581)

In 1932, K. Mahler and later J. K. Koksma in 1939 introduced related classifications of real and complex numbers in their studies of transcendence. There are many articles due to Bernik, Melnichuk, Sprindzhuk and Khintchine regarding the metric theory of diophantine approximation in the real case. In this paper the authors discuss the metric theory of diophantine approximation on complex manifolds. For the history of such investigations before 1967 see \textit{V. G. Sprindzhuk} [Mahler's problem in metric number theory, Minsk: Nauka i Tekhnika (1967; Zbl 0168.29504)]. Sprindzhuk proved the results for the curve \(\Gamma= \{(Z^m, Z^n): Z\in \mathbb{C}\}\) with \(m,n\) in \(\mathbb{N}\). The Hausdorff dimension for the curve \(\Gamma= \{(Z^m,Z^n,Z^l): Z\in \mathbb{C}\}\) was obtained by \textit{I. M. Morozova} [Vestsi Akad. Navuk Belarusi, Ser. Fiz.-Mat. Navuk 1997, No. 2, 22-25 (1997; Zbl 0896.11029)]. Metrical theorems regarding this curve and the curve \(\{(Z^m, Z^n, Z^l, Z^k): Z\in \mathbb{C}\}\) were obtained by \textit{V. I. Bernik} and \textit{N. V. Sakovich} [Dokl. Akad. Nauk Belarusi 38, No. 5, 10-13 (1994; Zbl 0831.11041)]. NEWLINENEWLINENEWLINELet \(\psi: \mathbb{R}\to \mathbb{R}^+\) be a function and let \(L_{m,n}(\psi)\) denote the set of points \(Z\in \mathbb{C}\) for which the inequality NEWLINE\[NEWLINE|a_2 Z^m+ a_1 Z^n+ a_0|< \psi^{1/2} (|a|)NEWLINE\]NEWLINE holds infinitely often where \(a= (a_0, a_1, a_2)\in \mathbb{Z}^3\) and \(|a|= \max|a_j|\), \(0\leq j\leq 2\). Let \(|A|\) denote the Lebesgue measure of a set \(A\), and \(B(a,r)\) the ball contained in \(A\) with center \(a\) and radius \(r\). It is proved here that NEWLINE\[NEWLINE|L_{m,n}(\psi)|\begin{cases} =0 &\text{if }\sum_{N=1}^\infty \psi(N)< \infty,\\ >0 &\text{if }\sum_{N=1}^\infty \psi(N)= \infty \text{ and \(\psi\) decreases} \end{cases}NEWLINE\]NEWLINE and \(|L_{2,1}(\psi)\cap B(0,r)|= \pi r^2\) when \(\sum \psi(N)\) diverges. For the convergence case the result was proved by the Sprindzhuk in 1967.NEWLINENEWLINEFor the entire collection see [Zbl 0932.00040].