## Diophantine exponents for mildly restricted approximation

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• Institut for Matematiske Fag
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.
Originalsprog Engelsk Arkiv foer Matematik 47 2 243-266 24 0004-2080 https://doi.org/10.1007/s11512-008-0074-0 Udgivet - 2009

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