Twisted Diophantine approximation on manifolds

2025/07/06 by Victor Beresnevich, Beresnevich, Victor, David Simmons +3 · 1 citation
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2507.04405

openalex publication_date 2025/07/06 · openalex created_date 2025/10/20 · openalex updated_date 2026/07/28

Abstract

In twisted Diophantine approximation, for a fixed m× n matrix \boldsymbolα one is interested in sets of vectors \boldsymbolβ∈\mathbb R^m such that the system of affine forms \mathbb Rⁿ \ni \mathbf q ↦ \boldsymbolα\mathbf q + \boldsymbolβ∈ \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.

Citations

• Rational points near manifolds and Khintchine theorem
• Borel-Cantelli, zero-one laws and inhomogeneous Duffin-Schaeffer
• Rational approximation with chosen numerators
• On a Kurzweil type theorem via ubiquity
• Extremal affine subspaces and Khintchine-Jarník type theorems
• Khintchine's theorem and Diophantine approximation on manifolds
• Inhomogeneous dual Diophantine approximation on affine subspaces
• Diophantine approximation in Kleinian groups: singular, extremal, and bad limit points
• Some manifolds of Khinchin type for convergence
• Metric Diophantine Approximation: aspects of recent work
• Badly approximable points in twisted Diophantine approximation and Hausdorff dimension
• Diophantine approximation on manifolds and the distribution of rational points: contributions to the convergence theory
• On Kurzweil's 0-1 Law in Inhomogeneous Diophantine Approximation
• A note on weighted badly approximable linear forms
• Quantitative shrinking target properties for rotations and interval exchanges
• Grids with dense values
• Inhomogeneous theory of dual Diophantine approximation on manifolds
• Modified Schmidt games and a conjecture of Margulis
• A solution to a problem of Cassels and Diophantine properties of cubic numbers
• On shrinking targets for Z^m actions on tori
• Classical metric Diophantine approximation revisited
• Measure theoretic laws for lim sup sets
• Extremal subspaces and their submanifolds
• Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions
• Metric Diophantine approximation: The Khintchine--Groshev theorem for non-degenerate manifolds
• Badly approximable systems of affine forms
• Flows on homogeneous spaces and Diophantine approximation on manifolds

Cited by

• Twisted Diophantine approximation for matrix transformations of tori

Related

• Khintchine's theorem and Diophantine approximation on manifolds
• Diophantine approximation on manifolds and the distribution of rational points: contributions to the convergence theory
• Non-planarity and metric Diophantine approximation for systems of linear forms
• Twisted Diophantine approximation for matrix transformations of tori
• Systems of small linear forms and Diophantine approximation on manifolds