Diophantine exponents for mildly restricted approximation
We are studying the Diophantine exponent μn,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 μn,l(α)=μ for μ≥n. Finally, letting wn 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 μn,1(α)≤...≤μn,n-1(α)≤wn(α) can be chosen to be arbitrary. 