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An application of a theorem of Singer
Published online by Cambridge University Press: 20 January 2009
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The authors have recently treated (2) the problem of finding subsets E of the real line , of type Fσ, such that E–E contains an interval and the k-fold vector sum (k)E is of measure zero. Positive results can be obtained, for all k, on the basis of a recent theorem of J. A. Haight (3), following earlier partial results (1), (4) for k ≦ 7; and indeed in these cases the problem has a solution with E a perfect set. An analogous problem, apparently in most respects subtler than the first, is the following. Do there exist finite regular Borel measures μ on
such that
is absolutely continuous (where
is the adjoint of μ) and the kth convolution power μk is singular? Both problems are of interest in the general context of elucidating the properties of the measure algebra
or, more generally, M(G) for locally compact abelian G. The second problem may be regarded as an attempt to provide (at least one aspect
of) a multiplicity theory for the first.
- Type
- Research Article
- Information
- Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 2 , September 1974 , pp. 119 - 123
- Copyright
- Copyright © Edinburgh Mathematical Society 1974
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