On Approximation constants for Liouville numbers

TL;DR

This paper studies how well Liouville numbers can be approximated simultaneously by rational numbers, explicitly determining approximation constants for a broad class including famous examples and Cantor set numbers.

Contribution

It explicitly computes all simultaneous approximation constants for certain classes of Liouville numbers, including well-known and Cantor set numbers.

Findings

Explicit formulas for approximation constants of Liouville numbers.

Includes classical Liouville number and Cantor set numbers.

Provides a comprehensive understanding of approximation quality for these numbers.

Abstract

We investigate some Diophantine approximation constants related to the simultaneous approximation of $(\zeta,\zeta^{2},\ldots,\zeta^{k})$ for Liouville numbers $\zeta$. For a certain class of Liouville numbers including the famous representative $\sum_{n\geq 1} 10^{-n!}$ and numbers in the Cantor set, we explicitly determine all approximation constants simultaneously for all $k\geq 1$.