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

We study two irrationality measure functions ${\psi }_{\alpha }^{[2]}(t)$ and ${\psi }_{\alpha }^{[2]\ast }(t)$ 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 ${\psi }_{\alpha }^{[2]\ast }(t)$ are associated with the numbers $\frac{1+\sqrt{5}}{2}$ and $e={\sum }_{n=0}^{\mathrm{\infty }}\frac{1}{n!}$.