Regularities in the distribution of special sequences. (English) Zbl 0531.10055

Hellekalek, Peter

J. Number Theory 18, 41-55 (1984).

For a given sequence \(\omega =(x(k))^{\infty}_{k=0}\) of points in \(U^ s=[0,1)^ s\), let \(S(\omega)\) denote the set of all subintervals \(J\) of \(U^ s\) with bounded discrepancy function, i.e. with \[ \sup_{n}| \sum^{n-1}_{k=0}(c_ J(x(k))-m(J))|<\infty, \] where \(c_ J\) is the characteristic function of \(J\) and \(m\) is the \(s\)-dimensional Lebesgue measure. W. M. Schmidt [Trans. Am. Math. Soc. 198, 1–22 (1974; Zbl 0278.10036)] showed that for any sequence \(\omega\) the set \(\{m(J): J\in S(\omega)\}\) is at most countable. The problem of determining \(S(\omega)\) explicitly has been studied for various special sequences \(\omega\).
In this paper the problem is considered for a class of generalized Halton sequences that are defined as follows. For \(1\leq i\leq s\) let \(q_ i=(q_ i(k))^{\infty}_{k=1}\) be a sequence of integers \(\geq 2\). Define \(p_ i(0)=1\) and \(p_ i(g)=q_ i(1)...q_ i(g)\) for all positive integers \(g\). It is assumed that \(\gcd(p_ i(k),p_ j(k))=1\) for \(1\leq i<j\leq s\) and all positive integers \(k\). Any nonnegative integer \(k\) has a unique \(q_ i\)-adic representation \(k=\sum^{\infty}_{j=0}a_ i(j)p_ i(j)\) with digits \(a_ i(j)\in \{0,1,...,q_ i(j+1)-1\}\) for \(1\leq i\leq s\). Define the radical-inverse function \(\Phi_ i\) to the base \(q_ i\) by \(\Phi_ i(k)=\sum^{\infty}_{j=0}a_ i(j)/p_ i(j+1).\) If \(\Phi(k)=(\Phi_ 1(k),\dots,\Phi_ s(k)),\) then \(\omega =(\Phi(k))^{\infty}_{k=0}\) is a generalized Halton sequence in \(U^ s\). If \(q_ i(j)=q_ i\) for all \(j\), one obtains a Halton sequence in \(U^ s\). A further generalization is obtained by applying permutations \(\sigma_ i(j)\) of the sets \(\{0,1,\dots,q_ i(j+1)-1\}\) to the digits \(a_ i(j)\) in the definition of \(\Phi_ i(k)\), thus arriving at a so-called \(\Sigma\)-\(q\)-adic sequence in \(U^ s.\)
If \(\omega\) is a \(\Sigma\)-\(q\)-adic sequence in \(U^ s\), then those \(J\in S(\omega)\) containing the origin are determined. For general elements \(J\) of \(S(\omega)\) a partial characterization is shown. In the case \(s=1\) the elements of \(S(\omega)\) are characterized completely. The proofs use methods from ergodic theory.

Reviewer: Harald Niederreiter

MSC:

11K38

Irregularities of distribution, discrepancy

Keywords:

sigma-\(q\)-adic sequence; discrepancy; generalized Halton sequences

Citations:

Zbl 0278.10036

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