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Hausdorff Measures and Sets of Uniqueness for Trigonometric Series

Dougherty, R. and Kechris, A. S. (1989) Hausdorff Measures and Sets of Uniqueness for Trigonometric Series. Proceedings of the American Mathematical Society, 105 (4). pp. 894-897. ISSN 0002-9939. doi:10.2307/2047049.

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We characterize the closed sets E in the unit circle T which have the property that, for some nondecreasing h: (0, ∞) →(0, ∞) with h(0+) = 0, all the Hausdorff h-measure 0 closed sets F ⊆ E are sets of uniqueness (for trigonometric series). In conjunction with Körner's result on the existence of Helson sets of multiplicity, this implies the existence of closed sets of multiplicity (M-sets) within which Hausdorff h-measure 0 implies uniqueness, for some h. This is contrasted with the case of closed sets of strict multiplicity (M_0-sets), where results of Ivashev-Musatov and Kaufman establish the opposite.

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Additional Information:© 1989 American Mathematical Society. Received by the editors April 21, 1988 and, in revised form, June 1, 1988. The second author was partially supported by NSF Grant DMS-8718847.
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MathSciNet ReviewMR0946633
Issue or Number:4
Classification Code:1980 Mathematics Subject Classification (1985 Revision). Primary 42A63, 43A46
Record Number:CaltechAUTHORS:20130524-113257744
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Official Citation:Hausdorff measures and sets of uniqueness for trigonometric series R. Dougherty and A. S. Kechris. Proc. Amer. Math. Soc. 105 (1989), 894-897
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:38667
Deposited By: Ruth Sustaita
Deposited On:24 May 2013 21:00
Last Modified:09 Nov 2021 23:39

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