\"Uber die Funktionen des Irrationalit\"atsma\ss es

TL;DR

This paper investigates two irrationality measure functions related to second-best approximations of real numbers, analyzing their spectra and identifying key elements associated with notable constants like the golden ratio and Euler's number.

Contribution

It introduces and studies two specific irrationality measure functions, revealing the structure of their spectra and identifying initial spectral elements linked to important mathematical constants.

Findings

The first two spectral elements of $ ext{spectra}( ext{function})$ are associated with $rac{1+ oot{2}\{5 ightrac{2}{2}$ and $e$.

The structure of the spectra for these functions is characterized and partially determined.

The functions relate to second-best approximations, providing new insights into Diophantine approximation spectra.

Abstract

We study two irrationality measure functions $\psi_\alpha^{[2]} (t) $ and $\psi_\alpha^{[2]*} (t)$ related to the "second best" approximations to a real numbers and prove some results on the structure of the corresponding Diophantine spectra. It happened that the first two elements of the spectrum for the function $\psi_\alpha^{[2]*} (t)$ are associated with the numbers $\frac{1+\sqrt{5}}{2}$ and $e =\sum_{n=0}^\infty \frac{1}{n!}$.