Non-degeneracy and Fibering with Applications to Diophantine Approximation

Source abstract

Motivated by applications in Diophantine approximation, we develop a new fibering mechanism for non-degenerate manifolds in $\mathbb{R}^n$, showing that they admit local decompositions into families of non-degenerate curves with uniform quantitative non-degeneracy properties. The manifolds are required to be $C^N$ with the essentially optimal bound $N\geq \max\lbrace n,l+1\rbrace$, where the manifold is $l$ non-degenerate. Furthermore, we show that this uniform quantitative local control can be globalised into a qualitative statement, so that almost every point of a non-degenerate manifold lies on a non-degenerate curve contained in the manifold. Our approach is perturbative and shows that the fibering structure arising in the analytic setting is stable under sufficiently small $C^N$ perturbations. The method provides a framework for extending fibering based results in Diophantine approximation from curves to manifolds. Such applications are considered in the paper.