On a problem related to "second" best approximations to a real number

TL;DR

This paper investigates the second-best approximation problem for irrational numbers, focusing on the Diophantine constant and spectrum, and computes the third element of the spectrum, extending previous work that identified the two largest elements.

Contribution

The paper extends prior research by calculating the third element of the spectrum of the Diophantine constant related to second-best approximations.

Findings

Calculated the third element of the spectrum  for the Diophantine constant.

Extended the known spectrum  by identifying its third largest element.

Abstract

For a given irrational number $\alpha$ one can define an irrationality measure function $\psi_{\alpha}^{[2]}(t) = \min\limits_{\substack{(q, p)\colon q, p \in\mathbb{Z}, 1\leqslant q\leqslant t, \\ (p, q) \neq (p_n, q_n) ~\forall n\in\mathbb{Z_{+}}}} |q\alpha -p|$, related to the second-best approximations to $\alpha$. In 2017 Moshchevitin studied the corresponding Diophantine constant $\mathfrak{k}(\alpha) = \liminf\limits_{t\to\infty} t \cdot\psi_{\alpha}^{[2]}(t)$ and the corresponding spectrum $\mathbb{L}_2 = \{ \lambda ~| ~\exists\alpha\in\mathbb{R\setminus Q} \colon \lambda = \mathfrak{k}(\alpha) \}$. In particular, he calculated two largest elements of the spectrum $\mathbb{L}_2$. In the present paper we calculate the value for the third element of the spectrum $\mathbb{L}_2$.