Removing monotonicity in Diophantine approximation

Simons Semester Continued Fractions, Fractals, Ergodic theory and Dynamics Thematic week on Diophantine approximation and Number Theory, 4–8.05.2026 Lecture by: Seongmin Kim “Removing monotonicity in Diophantine approximation” Khintchine’s theorem on metric Diophantine approximation establishes dichotomy for the measure of the set of ψ-approximable numbers. The problem of removing the monotonicity condition on ψ 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 ψ. This lecture was partially supported by the Simons Foundation grant (award no. SFI-MPS-T-Institutes-00010825) and from State Treasury funds as part of a task commissioned by the Minister of Science and Higher Education under the project “Organization of the Simons Semesters at the Banach Center - New Energies in 2026-2028” (agreement no. MNiSW/2025/DAP/491).