An \(F_\sigma\) semigroup of zero measure which contains a translate of every countable set. (English) Zbl 0614.28003
The present author proves the existence of an F-sigma set E in \(R^ n\) that contains a translate of every countable set, but whose every k-fold sum set has measure zero. The result is one of many similar examples originating with the 1942 example of S. Piccard [Sur les ensembles parfait (1942; Zbl 0027.20403)] of a set of real numbers whose sum set has zero measure but whose difference set contains an interval. The present work is also related to asymmetric Raikov systems and a recent paper by G. Brown and W. Moran [J. Lond. Math. Soc., II. Ser. 28, 531-542 (1983; Zbl 0545.43003)].
Reviewer: B.B.Wells jun
MSC:
| 28A75 | Length, area, volume, other geometric measure theory |
| 43A10 | Measure algebras on groups, semigroups, etc. |
| 28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |
| 28A12 | Contents, measures, outer measures, capacities |
Keywords:
asymmetric Raikov systemsReferences:
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