Twisted Diophantine approximation on manifolds

TL;DR

This paper develops a twisted Khintchine-type theory for Diophantine approximation on manifolds, where the inhomogeneous component ${\boldsymbol\beta}$ is restricted to a submanifold of ${\mathbb R}^m$ and the fixed matrix ${\boldsymbol\alpha}$ controls the affine forms. The authors introduce the framework of ${\boldsymbol\alpha}$-twisted Khintchine type for convergence/divergence over classes of approximation functions, and they establish sharp results for nondegenerate analytic manifolds, including convergence/divergence criteria tied to the exponent of irrationality of ${\boldsymbol\alpha}^T$, non-approximability conditions, and Hardy $L$-functions. They prove that under a transversality-type bound on ${\boldsymbol\alpha}^T$, nondegenerate analytic manifolds have twisted Khintchine-type convergence behaviour for doubling ψ, and under non-φ-approximability and Hardy $L$-function divergence assumptions, they obtain twisted Khintchine-type divergence with full measure on manifolds; they also address badly and very well approximable points, showing that for nonsingular ${\boldsymbol\alpha}$ the badly-approximable set is measure-zero on curves/analytic manifolds while very singular matrices yield full measures, and they provide absolute-winning dimension statements. A Dani correspondence for very singular matrices links Diophantine properties to flows on homogeneous spaces, enabling a comprehensive picture of twisted approximation on manifolds and resolving several subproblems in the twisted setting. Overall, the work significantly extends Khintchine-type twisted theory from ambient spaces to manifolds, with precise measure, dimension, and dynamical consequences.

Abstract

In twisted Diophantine approximation, for a fixed $m\times n$ matrix $\boldsymbolα$ one is interested in sets of vectors $\boldsymbolβ\in\mathbb R^m$ such that the system of affine forms $\mathbb R^n \ni \mathbf q \mapsto \boldsymbolα\mathbf q + \boldsymbolβ\in \mathbb R^m$ satisfies some given Diophantine condition. In this paper we introduce the notion of manifolds which are of $\boldsymbolα$-twisted Khintchine type for convergence or divergence. We provide sufficient conditions under which nondegenerate analytic manifolds exhibit this twisted Khintchine-type behaviour. Furthermore, we investigate the intersection properties of the sets of $\boldsymbolα$-twisted badly approximable and well approximable vectors with nondegenerate manifolds.

Theorems & Definitions (51)

• Theorem 1: Kurzweil, Kurzweil
• Remark 1.1
• Theorem 2: Khintchine type for convergence
• Remark 1.2
• Remark 1.3
• Theorem 3: Khintchine type for divergence
• Remark 1.4
• Theorem 4: Measure of badly ${\boldsymbol\alpha}$-approximable points, nonsingular case
• Theorem 5: Measure of badly ${\boldsymbol\alpha}$-approximable points, very singular case
• Remark 1.5