Arithmetic sums and products of infinite multiple zeta-star values

Multiple zeta-star values are variants of multiple zeta values which allow equality in the definition. Similar to the theory of continued fractions, every real number which is greater than $1$ can be realized as an unique infinite multiple zeta-star values in a natural way. In this paper, we investigate the arithmetic sums and products of infinite multiple zeta-star values with restricted indices. Moreover, inspired by the theory of continued fractions and Cantor set, we propose a series of conjectures concerning the algebraic points and arithmetic sums and products of infinite multiple zeta-star values with certain indices.