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On the distribution of s-dimensional Kronecker-sequences. (English) Zbl 0611.10033

The s-dimensional Kronecker sequence \(w:=(\{k\alpha _ 1\},...,\{k\alpha _ s\})_{k\in {\mathbb{N}}}\) is uniformly distributed in the s-dimensional unit cube \(I^ s\) if and only if \(1,\alpha _ 1,...,\alpha _ s\) are linearly independent over the rationals. The isotropic discrepancy \(J_ N\) of a sequence \((x_ n)_{n\in {\mathbb{N}}}\) in \(I^ s\) is defined by \[ J_ N:=\sup _{C}| \frac{\kappa (\{n\leq N | x_ n\in C\})}{N}- \mu (C)| \] where \(\mu\) is the s-dimensional Lebesgue measure and the supremum is taken over all convex subsets C of \(I^ s.\)
In this paper the isotropic discrepancy of s-dimensional Kronecker sequences is studied and partly best possible estimates are given. For example, two typical results are the following:
(i) \(\limsup _{N\to \infty}N^{1/s} \cdot J_ N<\infty\) if and only if \(L:=(\sum ^{s}_{j=1}m_ j\alpha _ j)-m\) is an extremal form.
(ii) If \(s=2\) then for all \((\alpha _ 1,\alpha _ 2)\in {\mathbb{R}}^ 2\) we have \(N^{1/2} \cdot J_ N\geq 0.0433..\). for infinitely many N.

MSC:

11K06 General theory of distribution modulo \(1\)
11K38 Irregularities of distribution, discrepancy
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