2008 | OriginalPaper | Buchkapitel
Orthogonality and Digit Shifts in the Classical Mean Squares Problem in Irregularities of Point Distribution
verfasst von : William W. L. Chen, Maxim M. Skriganov
Erschienen in: Diophantine Approximation
Verlag: Springer Vienna
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Suppose that
$$ \mathcal{A}_N $$
is a distribution of
N
> 1 points, not necessarily distinct, in the
n
-dimensional unit cube
U
n
=
[0,
l)
n
, where
n
≥ 2. We consider the L
2
-discrepancy
$$ \mathcal{L}_2 \left[ {\mathcal{A}_N } \right] = \left( {\int\limits_{U^n } {\left| {\mathcal{L}\left[ {\mathcal{A}_N ;Y} \right]} \right|} ^2 dY} \right)^{1/2} , $$
where for every
Y
= (y
1
,...,
y
n
) ∈
U
n
, the local discrepancy
$$ \mathcal{L}\left[ {\mathcal{A}_N ;Y} \right] $$
is given by
$$ \mathcal{L}\left[ {\mathcal{A}_N ;Y} \right] = \# \left( {\mathcal{A}_N \cap B_Y } \right) - N vol B_Y . $$
Here
$$ B_Y = \left[ {0,y_1 } \right) \times \ldots \times \left[ {0,y_n } \right) \subseteq U^n $$
is a rectangular box of volume vol
By =
y1...
y
n
, and #(
S
) denotes the number of points of a set
S
, counted with multiplicity.