Regular and ubiquitous systems for simultaneous Diophantine approximations

In this paper the Khintchine type theorem for polynomials in the divergence case is generalised to simultaneous approximations in $\mathbb{R}^k\times\mathbb{C}^l\times\mathbb{Q}_p^m$ and to approximations which incorporate a natural restriction on derivatives. The proof builds upon the construction of the optimal regular systems, the ubiquity technique and the construction of a system of close conjugate algebraic numbers.