Non-homogeneous analogue of Khintchine's theorem in divergence case for simultaneous approximations in different metrics


Natalia BUDARINA, Evgenii ZORIN

AMS Classification:

11J83, 11K60


Khintchine's type theorems, metric theory of Diophantine approximation, non-homogeneous Diophantine approximation


In this paper, we investigate the non-homogeneous analogue of Khintchine's theorem for simultaneous approximations. We prove that if the series \sum\limits_{r=1}^{\infty}\Psi(r) diverges, then the set of points (x,z,w)\in\RR\times\CC\Times\QQ_p which satisfy the inequalities |P(x)+d_1|<H^{-v_1}\Psi^{\lambda_1}(H), |P(z)+d_2|<H^{-v_2}\Psi^{\lambda_2}(H), |P(w)+d_3|<H^{-v_3}\Psi^{\lambda_3}(H) with d_1\in \mathbb{R} , d_2\in \mathbb{C}, d_3\in \mathbb{Z}_p, for infinitely many integer polynomials P has full measure.

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Vol. 4 (12), 2009