Diophantine exponents for mildly restricted approximation
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 sh...
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Published in | Arkiv för matematik Vol. 47; no. 2; pp. 243 - 266 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Dordrecht
Springer Netherlands
01.10.2009
|
Subjects | |
Online Access | Get full text |
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Summary: | 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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ISSN: | 0004-2080 1871-2487 |
DOI: | 10.1007/s11512-008-0074-0 |