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 ofSprind\u{z}uk's problem (1980) in the ring of adeles. We continueour investigation (2013), where the problem was proved at the pointsin $\mathbb{R}^2\times\mathbb{C}\times\mathbb{Q}_{p_1}$. We use themost precise form of {\it the essential and inessential domainsmethod} in metric theory of Diophantine approximation.
@article{310,
title = {The convergence part of a Knintchine-type theorem in the ring of adeles},
journal = {Tatra Mountains Mathematical Publications},
volume = {58},
year = {2014},
doi = {10.2478/tatra.v59i0.310},
language = {EN},
url = {http://dml.mathdoc.fr/item/310}
}
Kovalevskaya, Ella. The convergence part of a Knintchine-type theorem in the ring of adeles. Tatra Mountains Mathematical Publications, Tome 58 (2014) . doi : 10.2478/tatra.v59i0.310. http://gdmltest.u-ga.fr/item/310/