Hausdorff measures and sets of uniqueness for trigonometric series

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.