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

DOI:
https://doi.org/10.1090/S0002-9939-1989-0946633-0

MathSciNet review:
946633

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Abstract: We characterize the closed sets in the unit circle which have the property that, for some nondecreasing with , all the Hausdorff -measure 0 closed sets 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 (-sets) within which Hausdorff -measure 0 implies uniqueness, for some . This is contrasted with the case of closed sets of strict multiplicity ( -sets), where results of Ivashev-Musatov and Kaufman establish the opposite.

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DOI:
https://doi.org/10.1090/S0002-9939-1989-0946633-0

Article copyright:
© Copyright 1989
American Mathematical Society