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

Abstract

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, ω1, ω2) ∈ R × C × Qp1 × Qp2 , 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 × Qp1 . We use the most precise form of the essential and inessential domains method in metric theory of Diophantine approximation.