Talks

• Simons Semester: Continued Fractions, Fractals, Ergodic theory and Dynamics
  2026.05.07. IMPAN, Poland

  Abstract

  Title: Removing monotonicity in Diophantine approximation
  Abstract: Khintchine’s theorem on metric Diophantine approximation establishes a dichotomy for the measure of the set of $\psi$-approximable numbers. The problem of removing the monotonicity condition on $\psi$ from multi-dimensional, inhomogeneous variants of Khintchine’s theorem has been widely studied. In this talk, we will review previous results and discuss recent progress on $(n, m) = (1, 2)$ case under certain decay conditions on $\psi$.
• Rookies Pitch
  2025.09.23. Seoul National University, Korea Video

  Abstract

  Title: Khintchine’s theorem on Diophantine approximation
  Abstract: Diophantine approximation is the study of approximating real numbers by rational numbers. For example, one can ask whether a real number x is ψ-approximable; that is, whether there are infinitely many rationals p/q satisfying $\lvert x-p/q \rvert<ψ(q)/q$ for a given monotonic function ψ. A century ago, Khintchine discovered a remarkable dichotomy for the Lebesgue measure of the set of ψ-approximable numbers. Since then, Khintchine’s theorem has been extended in various directions, including inhomogeneous approximation and higher-dimensional generalizations. In this talk, I will introduce the Allen-Ramírez conjecture on removing the monotonicity condition from the inhomogeneous Khintchine-Groshev theorem and discuss a recent proof of the conjecture in the case (n,m)=(2,1).
• Diophantine approximation and related fields
  2025.06.11. University of York, UK

  Abstract

  Title: Inhomogeneous Khintchine—Groshev Theorem without Monotonicity
  Abstract: Khintchine’s theorem on metric Diophantine approximation has been developed in various directions over the past century. In particular, recent works by Yu, Allen and Ramírez, and Hauke have focused on removing the monotonicity condition from the inhomogeneous Khintchine–Groshev theorem. In this talk, we briefly review these developments and sketch a proof of the $(n, m) = (2, 1)$ case.
• 2025 KMS Spring Meeting [Poster]
  2025.04.25. KAIST, Korea

  Abstract

  Title: Inhomogeneous Khintchine-Groshev theorem without monotonicity
  Abstract: Diophantine approximation is the study of approximating a real number by rational numbers. For instance, given a monotonic function $\psi:\mathbb{N}→[0,1]$, one can ask whether a real number $x$ is $\psi$-approximable. While it is generally difficult to determine whether a specific number satisfies this condition, Khintchine established a dichotomy that almost every number is $\psi$-approximable, or almost none are. Groshev extended this result to the Diophantine approximation of $n \times m$ matrices, while Szusz and Schmidt further generalized it to inhomogeneous settings. After a long history of developments, Allen and Ramírez removed the monotonicity condition on $\psi$ in the case $nm\geq 3$ and conjectured that the result also holds for $nm=2$. In this work, we prove this conjecture for the $(n,m)=(2,1)$ case and obtain a partial result for the $(n,m)=(1,2)$ case. Our approach relies on analyzing the (quasi-)independence of certain measurable sets, with a gcd condition playing a key role in overcoming technical difficulties. ( Excellent Poster Award )
• KIAS Seminar
  2025.01.16. KIAS, Korea

  Abstract

  Title: Inhomogeneous Khintchine--Groshev Theorem without Monotonicity
  Abstract: Diophantine approximation is the study of approximating a real number by rational numbers. For instance, given a monotonic function ψ :N → [0,1], one can ask whether a real number x is ψ -approximable. While it is generally difficult to determine whether a specific number satisfies this condition, Khintchine established a dichotomy that almost every number is ψ -approximable, or almost none are. Groshev extended this result to the Diophantine approximation of n×m matrices, while Szusz and Schmidt further generalized it to inhomogeneous settings. After a long history of developments, Allen and Ramírez removed the monotonicity condition on ψ in the case nm ≥ 3 and conjectured that the result also holds for nm=2. In this talk, we briefly review earlier results and prove this conjecture for (n,m)=(2,1) case. The main strategy involves showing the (quasi-)independence of certain measurable sets, where the appropriate consideration of a gcd condition helps overcome key obstacles.
• Exchange Seminar
  2024.02.27. Seoul National University, Korea

  Abstract

  Title: On values of binary quadratic forms at integer points
  Abstract: Let Q be an indefinite binary quadratic form. The goal of this talk is to introduce estimates for the number of integral solutions in large balls, of the form $\lvert Q(x,y)\rvert < ε$, obtained by Choudhuri and Dani (2015). The proof consists of dynamics, hyperbolic geometry and negative Hurwitz continued fraction.