Uniform partitions of an interval

Author:
Vladimir Drobot

Journal:
Trans. Amer. Math. Soc. **268** (1981), 151-160

MSC:
Primary 10K30; Secondary 10K05

MathSciNet review:
628451

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Abstract: Let be a sequence of numbers in ; for each let be the lengths of the intervals resulting from partitioning of by . For put ; the paper investigates the behavior of as for various sequences . Theorem 1. *If* *for an irrational* , *then* . *However* *if and only if the partial quotients of* *are bounded* (*in the continued fraction expansion of* ). Theorem 2 gives the exact values for and when . Theorem 3. *If* *are random variables, uniformly distributed on* , *then* *almost surely*.

**[1]**J. L. Doob,*Stochastic processes*, John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. MR**0058896****[2]**V. Drobot,*Approximation of curves by polygons*, J. Number Theory**17**(1983), no. 3, 366–374. MR**724535**, 10.1016/0022-314X(83)90054-9**[3]**I. S. Gradshtein and I. M. Ryzhik,*Tables of integrals, sums and products*, Moscow, 1971.**[4]**G. H. Hardy and E. M. Wright,*An introduction to the theory of numbers*, Clarendon Press, Oxford, 1960.**[5]**Samuel Karlin,*A first course in stochastic processes*, Academic Press, New York-London, 1966. MR**0208657****[6]**L. Kuipers and H. Niederreiter,*Uniform distribution of sequences*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR**0419394****[7]**Noel B. Slater,*Gaps and steps for the sequence 𝑛𝜃𝑚𝑜𝑑1*, Proc. Cambridge Philos. Soc.**63**(1967), 1115–1123. MR**0217019**

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1981-0628451-3

Article copyright:
© Copyright 1981
American Mathematical Society