Diophantine approximation of multiple zeta-star values

The set of multiple zeta-star values is a countable dense subset of the half line $(1,+\mathrm{\infty })$. In this paper, we establish some classical Diophantine type results for the set of multiple zeta-star values. Firstly, we give a criterion to determine whether a number is a multiple zeta-star value. Secondly, we establish the zero-one law for the set of multiple zeta-star value. Lastly, we propose a conjecture for the set of multiple zeta-star values, which strengthens the original zero-one law.