Metrical theorems on restricted Diophantine approximations to points on a curve.
In a previous paper [J. Number Theory 29, 364--375 (1988; Zbl 0655.10049)], the first author proved that for almost all \(\alpha>0\) there are infinitely many primes \(p,q\) such that \(| p\alpha-q| <\psi(p)\), if \(\psi\) satisfies a mild hypothesis and the sum over primes \(\sum\psi(p)(\log p)^{-1}\) diverges. This has been generalized to higher dimensions by the second author [Acta Arith. 99, 205--225 (2001; Zbl 1013.11040)]. In the present paper the authors consider the much more difficult problem of obtaining such approximations to almost all points on a curve in \(\mathbb R^2\). Their main result is: Let \(\varepsilon>0\), \(\tau>1\), and \(\mu=\frac{3}{16}\). Then for almost all \(\alpha>0\) there are infinitely many primes \(p,q,r\) such that \(0<p\alpha-r<p^{-\mu+\varepsilon}\), \(0<p\alpha^\tau-q<p^{-\mu+\varepsilon}.\)
scientific article; zbMATH DE number 1891427
scientific article