Title: Lagrange and Markov Spectra
Abstract: The Lagrange spectrum and the Markov spectrum arise from optimal Diophantine approximation and the study of indefinite binary quadratic forms, respectively. It is well known that these two spectra are deeply related. Using Perron’s formula, we interpret both spectra dynamically through the shift on bi-infinite continued fraction sequences. This viewpoint explains the structural properties of both spectra.
In this survey talk, we discuss classical results of Markov, Perron, Hall, and Freiman, including the existence of isolated points in the complement of the Lagrange spectrum inside the Markov spectrum. Finally, we will discuss generalizations to Fuchsian groups and the spectra of the Hecke group.
Title: Best approximation over function fields revealed by Hankel matrices
Abstract: In function fields, rational approximation is measured by how many leading terms of a Laurent series can be matched by a rational function with a bounded-degree denominator. This talk explains how Hankel matrices built from the coefficient sequence provide a concrete and surprisingly sharp way to detect when “best denominators” occur and when unusually strong approximations are possible. The key message is that patterns of vanishing and non-vanishing Hankel minors act as a clean algebraic fingerprint of approximation quality: vanishing signals the existence of exceptionally good Padé-type approximants at specific degrees, while non-vanishing imposes rigidity and forces best approximations to appear only at predictable scales. We will connect this Hankel viewpoint to geometric intuition—record points and visible vectors in a sheared lattice, and successive-minima pictures in expanding boxes—emphasizing what becomes simpler in the non-Archimedean setting.
Title: Arithmetical properties of real numbers with certain β-expansions
Abstract: Let $k \geq 2$ be an integer and $\beta$ be a Pisot number or a Salem number. Define $\gamma(k;\beta) := \sum_{n=0}^{\infty} \beta^{-n^k}$. In 1957, Erd\H{o}s investigated the arithmetical properties of values of power series. On the other hand, Bailey, Borwein, Crandall and Pomerance (2004) gave the lower bound for the density of nonzero digits in the binary expansion of algebraic numbers. This result was generalized to the case of $\beta$-expansions. In particular, $\gamma(k;\beta)$ is either a transcendental number or an algebraic number of degree at least $k$. Moreover, for any integer $b \geq 2$, Murakami and Tachiya (2024) showed, as a special case of their result, that $1, \gamma(k,b), (k=2, 3, \cdots)$ are linearly independent over $\mathbb{Q}$. Kudo (2025) generalized this result from $b$ to $\beta$. However the algebraic independence of the values $\gamma(k;\beta)$ is still unknown. In this talk, we give the linear independence of powers of such numbers $\gamma(k_1;\beta),\cdots,\gamma(k_r;\beta)$ with degree at most $D$ for given positive integers $r$ and $D$ under certain assumptions.
Contact: Dong Han Kim (kim2010@dgu.ac.kr)