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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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.
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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