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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Hausdorff measures and sets of uniqueness for trigonometric series


Authors: R. Dougherty and A. S. Kechris
Journal: Proc. Amer. Math. Soc. 105 (1989), 894-897
MSC: Primary 42A63; Secondary 28A75, 43A46
MathSciNet review: 946633
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Abstract: We characterize the closed sets $ E$ in the unit circle $ {\mathbf{T}}$ which have the property that, for some nondecreasing $ h:\left( {0,\infty } \right) \to \left( {0,\infty } \right)$ with $ h\left( {0 + } \right) = 0$, all the Hausdorff $ h$-measure 0 closed sets $ F \subseteq 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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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0946633-0
PII: S 0002-9939(1989)0946633-0
Article copyright: © Copyright 1989 American Mathematical Society