The convergence part of a Knintchine-type theorem in the ring of adeles

Abstract

We prove the convergence part of a Khintchine-type theorem for simultaneous Diophantine approximation of zero by values of integral polynomials at the points

$$

(x,z,\omega_1,\omega_2)

\in

\mathbb{R}\times\mathbb{C}\times\mathbb{Q}_{p_1}

\times

\mathbb{Q}_{p_2},

$$

where $p_1\neq p_2$ are primes. It is a generalization of
Sprind\u{z}uk's problem (1980) in the ring of adeles. We continue
our investigation (2013), where the problem was proved at the points
in $\mathbb{R}^2\times\mathbb{C}\times\mathbb{Q}_{p_1}$. We use the
most precise form of {\it the essential and inessential domains
method} in metric theory of Diophantine approximation.