Mesure de Hausdorff des complementaires desU _ε de zygmund

Abstract

A. Zygmund originally introduced the notion of sets of unicityU _ε, and showed that they differ from classicalU-sets in that they can have positive measure. He then asked if they could be of full measure. J. P. Kahame and Y. Katznelson proved recently that there wereU _ε of full measure. The object of this paper is to show that, in terms of Hausdorff measure, one cannot go beyond that result, for a general sequence ε. In the case of a given sequence ε, and a given Hausdorff determining functionh, it gives a simple test for determining the existence ofU _ε with a complement of zero Hausdorff measure. In this paper the proof of the main known results concerning the measure ofU _ε sets is also reproduced.