On the spectrum of Diophantine approximation constants

TL;DR

This paper explores the joint spectrum of Diophantine approximation constants, establishing new connections and extending known results to higher dimensions, including explicit constructions of numbers with prescribed approximation properties.

Contribution

It introduces a new connection on the joint spectrum of approximation constants, extending Bugeaud's results to arbitrary dimensions and providing explicit constructions within the Cantor set.

Findings

Established a connection on the joint spectrum of approximation constants.

Extended results on the individual spectrum to arbitrary dimension $k$.

Provided explicit constructions of numbers with specific approximation constants.

Abstract

The approximation constant $\lambda_{k}(\zeta)$ is defined as the supremum of real $\eta$ such that $\Vert \zeta^{j}x\Vert\leq x^{-\eta}$ for $1\leq j\leq k$ has infinitely many integer solutions $x$. Here $\Vert.\Vert$ denotes the distance to the closest integer. We establish a connection on the joint spectrum $(\lambda_{1}(\zeta),\lambda_{2}(\zeta),\ldots)$ which will lead to various improvements of known results on the individual spectrum of the approximation constants $\lambda_{k}(\zeta)$ as well. In particular, this extends a result by Bugeaud to the case of arbitrary dimension $k$. Concretely, given $k\geq 1$ and $\lambda\geq 1$, we infer {\em explicit} constructions of $\zeta$ in the Cantor set with $\lambda_{k}(\zeta)=\lambda$.