Hausdorff Measures and Sets of Uniqueness for Trigonometric Series

Abstract

Abstract.We characterize the closed sets E in the unit circle T which have the property that, for some nondecreasing h: (0,oo) -» (0,oo) with h(0+) = 0 , all the Hausdorff /¡-measure 0 closed sets F C 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-sels) within which Hausdorff /i-measure 0 implies uniqueness, for some h . This is contrasted with the case of closed sets of strict multiplicity ( A/o-sets), where results of Ivashev-Musatov and Kaufman establish the opposite.A Hausdorff determining function is any function h : (0 , oo) -► (0, oo) which is nondecreasing and satisfies h(0+) = 0. A subset E of T (the unit circle, viewed here as R/2nZ) has Hausdorff h-measure 0 if for every e > 0 there is a sequence {In} of open intervals (arcs) in T with E ç \Jn En and Z)^(l^sl) < e • (Here |/J = (arc length of I"/2n.) It is a well-known theorem of Ivashev-Musatov [2] that metric thinness in the form of Hausdorff «-measure 0 cannot imply uniqueness for trigonometric series. More precisely, for any « as above there is a closed set E of Hausdorff «-measure 0 which is of restricted multiplicity (we will give a review of terminology below). Kaufman [3; see also 4, VIII.3.3] extended this result by proving that, for any « as above, any closed set of strict multiplicity has a closed subset of Hausdorff «-measure 0 which is still of restricted multiplicity. Later Kaufman asked (in a private conversation) whether the same holds for ordinary multiplicity. The main result of this paper gives a negative answer to Kaufman's question, and provides a characterization of those closed sets which, for some « , contain no closed subsets of Hausdorff «-measure 0 which are of multiplicity.We now give a brief review of the notation, terminology, and results we need; this material all appears in Kechris-Louveau [4]. Let K(T) be the collection of closed subsets of T. Denote by U the collection of sets E e K(T) which are of uniqueness (i.e., every trigonometric series converging to 0 on T\E is identically 0), and let M = K(T)\U (the collection of closed sets of multiplicity). This is