Kronecker sets and metric properties of $M\sb{0}$-sets

Abstract

A method for constructing both sets of multiplicity and Kronecker sets within a given set of multiplicity is derived from the work of Ivashev-Musatov; it is shown that the Hausdorff measures and other measures are essentially distinct. Finally, an improvement of a theorem of Salem is obtained, using PyateckilShapiro's theorem on non-M sets. 1. The class of complex Borel measures , on the real axis, such that fi(u)-0 as Iul->oo, is denoted R, and a closed set is called Mo if it supports some measure ,u#O of class R. With a measure , the entire space L1(1) is contained in R [7, I, p. 143], so that we mostly study probability measures in R. The most striking examples of non-MO sets are the Kronecker sets: A compact linear set E is a K-set if each continuous function on E of modulus 1 admits uniform approximation on E by characters Z(x)= e2Ur (-oo O, then E1r(E2+x) is MO for an x-set of positive Lebesgue measure. A sharp converse is true. THEOREM 1. Let F be a closed set of Lebesgue measure in(F)=O. Then there exists an MO-set E so that (F+x) r)E is a K-set for every real x. Theorem 1 is a consequence of a more general assertion; for closed sets F and numbers r>O let m(F, r)=m{x: dist(x, F) g(r)for all r<rO(F). Theorem 1' is derived from a very general theorem in which Lebesgue measure plays no special role. THEOREM 2. Let It be a probability of class R and of closedl support S; let (Tk)Y be a sequence of closed sets w ith lim y (T)=0. Then there exists an Received by the editors January 19, 1972. AMS (MOS) subject classJifcations (1969). Primary 4250, 4252; Secondary 4256.