On a problem related to "second" best approximations to a real number
For a given irrational number $α$ one can define an irrationality measure function $ψ_α^{[2]}(t) = \min\limits_{\substack{(q, p)\colon q, p \in\mathbb{Z}, 1\leqslant q\leqslant t, \\ (p, q) \neq (p_n, q_n) ~\forall n\in\mathbb{Z_{+}}}} |qα-p|$, related to the second-best approximations to $α$. In 2017 Moshchevitin studied the corresponding Diophantine constant $\mathfrak{k}(α) = \liminf\limits_{t\to\infty} t \cdotψ_α^{[2]}(t)$ and the corresponding spectrum $\mathbb{L}_2 = \{ λ~| ~\existsα\in\mathbb{R\setminus Q} \colon λ= \mathfrak{k}(α) \}$. 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 $\mathbb{L}_2$.
math.NT↗