Analog of Khinchin's theorem in case of divergence in the fields of real, complex and $p$-adic numbers

A. S. Kudin ; A. V. Lunevich

Trudy Instituta matematiki, Tome 23 (2015) no. 1, pp. 76-83

Résumé

In this paper it is proved that if a positive function $\mathit\Psi$ is monotonically decreasing and a series $\sum_{r=1}^\infty\mathit\Psi(r)$ diverges, then the set of points $(x,z,\omega)\in\mathbb{R}\times\mathbb{C}\times\mathbb{Q}_p$ for which there are infinitely many polynomials, such that the inequalities are satisfied $$ |P(x)|^{-v_1}\mathit\Psi^{\lambda_1}(H), \quad |P(z)|^{-v_2}\mathit\Psi^{\lambda_2}(H), \quad |P(\omega)|_p^{-v_3}\mathit\Psi^{\lambda_3}(H) $$ (where is $v_1+2v_2+v_3=n-3,$ $\lambda_1+2\lambda_2+\lambda_3=1,$ $n$ — polynomial degree, $v_i,\lambda_i>0,$ $i=1,2,3$), has full measure.