Diophantine approximation and special Liouville numbers

TL;DR

This paper develops methods to determine simultaneous approximation constants for certain well approximable numbers, linking their s-adic expansions to approximation properties, and constructs explicit examples with prescribed approximation features.

Contribution

It introduces new techniques connecting s-adic expansions to approximation constants and provides explicit constructions of numbers with specific approximation characteristics.

Findings

Established methods for simultaneous approximation constants

Linked s-adic expansions to approximation properties

Constructed explicit examples with prescribed approximation features

Abstract

This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers $\zeta_{1},\zeta_{2},...,\zeta_{k}$. The approach relies on results on the connection between the set of all $s$-adic expansions ($s\geq 2$) of $\zeta_{1},\zeta_{2},...,\zeta_{k}$ and their associated approximation constants. As an application, explicit construction of real numbers $\zeta_{1},\zeta_{2},...,\zeta_{k}$ with prescribed approximation properties are deduced and illustrated by Matlab plots.