Diophantine exponents for mildly restricted approximation

TL;DR

This paper investigates the range of Diophantine exponents for vectors approximated within certain geometric constraints, characterizing their possible values, measure-theoretic properties, and the flexibility of their gaps.

Contribution

It introduces a comprehensive analysis of the Diophantine exponent fa_{n,l} for restricted approximation, including its possible values, Hausdorff dimension, and gap properties.

Findings

Exponent fa_{n,l} takes all values in [l+1, {n}].

Almost all vectors fa have exponent {n}.

Gaps in the sequence of exponents can be arbitrarily chosen.

Abstract

We are studying the Diophantine exponent \mu_{n,l}$ defined for integers 1 \leq l < n and a vector \alpha \in \mathbb{R}^n by letting \mu_{n,l} = \sup{\mu \geq 0: 0 < ||x \cdot \alpha|| < H(x)^{-\mu} for infinitely many x \in C_{n,l} \cap \mathbb{Z}^n}, where \cdot is the scalar product and || . || denotes the distance to the nearest integer and C_{n,l} is the generalised cone consisting of all vectors with the height attained among the first l coordinates. We show that the exponent takes all values in the interval [l+1, \infty), with the value n attained for almost all \alpha. We calculate the Hausdorff dimension of the set of vectors \alpha with \mu_{n,l} (\alpha) = \mu for \mu \geq n. Finally, letting w_n denote the exponent obtained by removing the restrictions on x, we show that there are vectors \alpha for which the gaps in the increasing sequence \mu_{n,1} (\alpha) \leq ... \leq \mu_{n,n-1} (\alpha) \leq w_n (\alpha) can be chosen to be arbitrary.