Diophantine exponents for mildly restricted approximation

Abstract

We are studying the Diophantine exponent μ_{n, l} defined for integers 1≤l< n and a vector α∈ℝⁿ by letting $\mu_{n,l}=\sup\{\mu\geq0: 0 < \Vert\underline{x}\cdot\alpha\Vert<H(\underline{x})^{-\mu}\ \text{for infinitely many}\ \underline{x}\in\mathcal{C}_{n,l}\cap\mathbb{Z}^n\},$ where $\cdot$ is the scalar product, $\|\cdot\|$ denotes the distance to the nearest integer and $\mathcal{C}_{n,l}$ 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,∞), with the value n attained for almost all α. We calculate the Hausdorff dimension of the set of vectors α with μ_{n, l}(α)=μ for μ≥n. Finally, letting w_n denote the exponent obtained by removing the restrictions on $\underline{x}$, we show that there are vectors α for which the gaps in the increasing sequence μ_n,1(α)≤...≤μ_{n, n-1}(α)≤w_n(α) can be chosen to be arbitrary.