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**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 S_{1},S_{2} are subsets such that for each i = 1,2 there is an infinite subset R_{i} of R so that the sets rS_{i}(r in R_{i}) are pairwise disjoint. Is it true that S_{1}\cup S_{2} 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 S_{1},S_{2},...,S_{k} 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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