Abstract: We report on two recent works concerning the Lagrange spectrum and its variants. Moshchevitin introduced the second Lagrange spectra $L_2$ and $L_2^*$ by considering the problem of approximating an irrational number by rational numbers that are not convergents of its continued fraction expansion. In joint work with H. Cheng, T. Vasconcelos, and Gugu, we prove that the function $d_2(t)=HD(L_2\cap (-\infty,t))$ is continuous, whereas $d_2^(t)=HD(L_2^*\cap (-\infty,t))$ is discontinuous and assumes only the values 0 and 1.
In a separate collaboration with Zhe Cao and Gugu, we completely characterize the set of irrational numbers $x$ for which the inequality $\left|x-\frac{p}{q}\right|<\frac{1}{3q^2}$ has only finitely many rational solutions. In particular, for every algebraic real number $x$ of degree at least 3, there exist infinitely many rational numbers $\frac{p}{q}$ such that $\left|x-\frac{p}{q}\right|<\frac{1}{3q^2}$.
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