Diophantine exponents for mildly restricted approximation

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  • Department of Mathematical Sciences
We are studying the Diophantine exponent μ n,l defined for integers 1≤l<n and a vector α∈ℝ n 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.
Original languageEnglish
JournalArkiv foer Matematik
Volume47
Issue number2
Pages (from-to)243-266
Number of pages24
ISSN0004-2080
DOIs
StatePublished - 2009

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