On the spectrum of Diophantine approximation constants

Abstract

The approximation constant ${\lambda }_k(\zeta )$ is defined as the supremum of real $\eta$ such that $\Vert {\zeta }^{j}x\Vert \le x^{-\eta }$ for $1\le j\le k$ has infinitely many integer solutions $x$. Here $\Vert .\Vert$ denotes the distance to the closest integer. We establish a connection on the joint spectrum $({\lambda }_1(\zeta ),{\lambda }_2(\zeta ),\dots )$ 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 $k\ge 1$ and $\lambda \ge 1$, we infer {\em explicit} constructions of $\zeta$ in the Cantor set with ${\lambda }_k(\zeta )=\lambda$.

21 pages. The false citation of the right transference inequality in (56) was corrrected

References in corpus (3)

• Diophantine approximation in small degree
• Exponents of Diophantine approximation
• On Approximation constants for Liouville numbers

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• Going-up theorems for simultaneous Diophantine approximation
• On Approximation constants for Liouville numbers
• Khintchine's theorem and Diophantine approximation on manifolds
• Rational approximation to algebraic varieties and a new exponent of simultaneous approximation
• Counting rational points close to $p$-adic integers and applications in Diophantine approximation