Hausdorff measures and sets of uniqueness for trigonometric series
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- by R. Dougherty and A. S. Kechris PDF
- Proc. Amer. Math. Soc. 105 (1989), 894-897 Request permission
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.References
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Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 105 (1989), 894-897
- MSC: Primary 42A63; Secondary 28A75, 43A46
- DOI: https://doi.org/10.1090/S0002-9939-1989-0946633-0
- MathSciNet review: 946633