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On a class of uniformly distributed sequences. (English) Zbl 0735.11034

A sequence \((x_ n)\), \(x_ i\in[0,1]\) is called uniformly distributed in blocks of length \(a_ n\) if \[ \lim_{N\to\infty} {\text{Card}\{n\leq a_ 1+a_ 2+\cdots+a_ N:x_ n\in I\}\over a_ 1+a_ 2+\cdots+a_ N}=| I| \] for all intervals \(I\subseteq[0,1]\). For such sequences which are blockwise monotone i.e. \(x_ n<x_ m\) for \(\sum_{i=1}^ ka_ i<n<m\leq\sum_{i=1}^{k+1}a_ i\) it is shown that \[ \lim_{n\to\infty}{a_{n+1}\over a_ 1+a_ 2+\cdots+a_ n}=0 \] is necessary and sufficient for \((x_ n)\) to be uniformly distributed. Sequences \((a_ n)\) satisfying this criterion are investigated and metrical and topological properties of the set of sequences \((a_ 1,a_ 1+a_ 2,\ldots,\sum_{i=1}^ na_ i,\ldots)\) with \((a_ i)\) satisfying the above condition in the set of increasing sequences are deduced.

MSC:

11K31 Special sequences
11K06 General theory of distribution modulo \(1\)
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References:

[1] ABEL N. H.: Note sur un mémoire de M. L. Olivier, ayant pour title ”Remarques sur les séries infinies et leur convergenee”. J. Reine Angew. Math. 3, 1828, 49-52.
[2] de BRUIJN N. G., POST K. A.: A remark on uniformly distributed sequences and Riemann integrability. Indagationes Math. 30, 1968, 149-150. · Zbl 0169.38401
[3] KNAPOWSKI S.: Über ein Problem der Gleichverteilungstheorie. Colloq. Math. 5, 1958, 8-10. · Zbl 0083.04401
[4] KNOPP K.: Theory and Applications of Infinite Series. Blackie and Son, London-Glasgow 1957. · Zbl 0842.40001
[5] KUIPERS L., NIEDERREITER H.: Uniform Distribution of Sequences. J. Wiley and Sons, New York-London-Sydney-Toronto 1974. · Zbl 0281.10001
[6] KURATOWSKI K.: Wstep do teorii mnogosci i topologii. PWN, Warszawa 1955.
[7] NIEDERREITER H.: Distribution mod 1 of monotone sequences. Indag. Math. 46, 1984, 315-327. · Zbl 0549.10038
[8] OSTMANN H. H.: Additive Zahlentheorie. Springer Verlag, Berlin-Göttingen-Heideiberg 1956. · Zbl 0072.03102
[9] PÓLYA G., SZEGÖ G.: Aufgaben und Lehrsätze aus der Analysis. Springer Verlag, Berlin 1964. · Zbl 0122.29704
[10] ŠALÁT T.: O mere Chausdorfa linejnych množestv. Czechoslovak Math. J. 11 (86), 1961, 24 -56.
[11] SIVAŠINSKIJ I. Ch.: Neravenstva v zadačach. Nauka, Moskva 1967.
[12] VOLKMANN B.: Über Hausdorffsche Dimensionen von Mengen, die durch Ziffereigensehaften charakterisiert sind II. Math. Z. 59, 1953, 247-254. · Zbl 0051.29701 · doi:10.1007/BF01180255
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