Abstract
We are studying the Diophantine exponent μ n,l defined for integers 1≤l<n and a vector α∈ℝ n by letting
where 
is the scalar product, 
denotes the distance to the nearest integer and 
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 
, 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.
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is the scalar product, denotes the distance to the nearest integer and 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 , 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.| Originalsprog | Engelsk |
|---|---|
| Tidsskrift | Arkiv foer Matematik |
| Vol/bind | 47 |
| Nummer | 2 |
| Sider (fra-til) | 243-266 |
| Antal sider | 24 |
| ISSN | 0004-2080 |
| DOI | |
| Status | Udgivet - 2009 |