Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  472.28009
Autor:  Mauldin, R.Daniel; Erdös, Paul
Title:  Rotations of the circle. (In English)
Source:  Measure theory, Proc. Conf., Oberwolfach 1979, Lect. Notes Math. 794, 53- 56 (1980).
Review:  [For the entire collection see Zbl 418.00006.]
The paper is addressed to the following questions: Let T be the unit circle and suppose S1,S2 are subsets such that for each i = 1,2 there is an infinite subset Ri of R so that the sets rSi(r in Ri) are pairwise disjoint. Is it true that S1\cup S2 has inner Lebesgue measure 0? The answer is yes, and two proofs are given. The first proof is a simple application of the amenability of T as discrete group, and extends to arbitrary locally compact groups which are amenable as discrete groups, and to k sets S1,S2,...,Sk instead of two. The second proof is more elementary and based on a counting argument. Several problems are given which arise from consideration of these proofs.
Reviewer:  W.Moran
Classif.:  * 28C10 Set functions and measures on topological groups
28D05 Measure-preserving transformations
43A05 Measures on groups, etc.
28A99 Classical measure theory
Keywords:  translation; amenable group; unit circle; inner Lebesgue measure 0
Citations:  Zbl.418.00006

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