The Convergence Part of a Knintchine-Type Theorem in the Ring of Adeles

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, ω₁, ω₂) ∈ R × C × Q_p1 × Q_p2 ,

where p₁ ≠ p₂ are primes. It is a generalization of Sprindžuk’s problem (1980) in the ring of adeles. We continue our investigation (2013), where the problem was proved at the points in R² × C × Q_p1 . We use the most precise form of the essential and inessential domains method in metric theory of Diophantine approximation.