Singular vectors on manifolds and fractals

TL;DR

This paper extends Khintchine's method to construct totally irrational singular vectors and linear forms with large Diophantine exponents on certain manifolds, including analytic submanifolds of dimension at least 2.

Contribution

It generalizes the construction of singular vectors to broader subsets of Euclidean space, especially on analytic submanifolds of dimension ≥ 2.

Findings

Existence of totally irrational vectors with large uniform Diophantine exponents on specific manifolds.

Construction method applicable to analytic submanifolds not contained in rational affine subspaces.

Extends classical Diophantine approximation results to new geometric contexts.

Abstract

We generalize Khintchine's method of constructing totally irrational singular vectors and linear forms. The main result of the paper shows existence of totally irrational vectors and linear forms with large uniform Diophantine exponents on certain subsets of $\mathbb{R}^n$, in particular on any analytic submanifold of $\mathbb{R}^n$ of dimension $\ge 2$ which is not contained in a proper rational affine subspace.