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Published April 1989 | Published
Journal Article Open

Hausdorff Measures and Sets of Uniqueness for Trigonometric Series

Abstract

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.

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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August 19, 2023
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