Inhomogeneous Khintchine-Groshev theorem without monotonicity

2024/11/12 by Seongmin Kim, Kim, Seongmin · 1 citation
Mathematics · #11J20 #11J83 #11K60 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Functional Equations Stability Results #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2411.07932

openalex publication_date 2024/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Khintchine-Groshev theorem in Diophantine approximation theory says that there is a dichotomy of the Lebesgue measure of sets of ψ-approximable numbers, given a monotonic function ψ. Allen and Ramírez removed the monotonicity condition from the inhomogeneous Khintchine-Groshev theorem for cases with nm≥3 and conjectured that it also holds for nm=2. In this paper, we prove this conjecture in the case of (n,m)=(2,1). We also prove it for the case of (n,m)=(1,2) with a rational inhomogeneous parameter.

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