Simultaneous diophantine approximation in $\mathbb{R}^2 × \mathbb{C} × \mathbb{Q}_p$

Abstract

An analogue of the convergence part of Khintchine's theorem (1924) for simultaneous approximation of integral polynomials at the points $(x_1,x_2,z,w)\in\mathbb{R}^2\times\mathbb{C}\times\mathbb{Q}_p$ is proved. It is a solution of the more general problem than Sprind\u{z}uk's problem (1980) in the ring of adeles. We use a new form of the essential and nonessential domains method in metric theory of Diophantine approximation.