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This survey paper presents some old and new results in Diophantine approximations. Some of these results improve Erdos' results on~irrationality. The results in irrationality, transcendence and linear independence of infinite series and infinite products are put together with idea of irrational sequences and expressible sets.

The paper deals with several criteria for the transcendence of infinite products of the form ${\prod }_{n=1}^{\infty }\left[b_n{\alpha }^{a_n}\right]/b_n{\alpha }^{a_n}$ where $\alpha >1$ is a positive algebraic number having a conjugate ${\alpha }^{*}$ such that $\alpha \ne |{\alpha }^{*}|>1$, ${\left\{a_n\right\}}_{n=1}^{\infty }$ and ${\left\{b_n\right\}}_{n=1}^{\infty }$ are two sequences of positive integers with some specific conditions. The proofs are based on the recent theorem of Corvaja and Zannier which relies on the Subspace Theorem (P. Corvaja, U. Zannier: On the rational approximation to the powers of an algebraic number: solution of two problems of Mahler and Mendès France, Acta...