M-Sets and Measures
Robert Bruce Kaufman · Annals of Mathematics · 1992
The measures pt to be investigated are defined on the torus T and satisfy lim suplii(n)I > 0. Lyons [8] recently proved the striking theorem that every measure ,t of this type must have positive mass in some closed set of type UO. Later Lyons asked whether UO could be improved to U and provided [9] ingenious counter-examples for the classical examples of U-sets. (Definitions and elementary results in harmonic analysis may be found in works cited a few lines below.) We prove the following: