EXISTENCE OF LARGE ε-KRONECKER AND FZI0(U) SETS IN DISCRETE ABELIAN GROUPS

Let G be a compact abelian group with dual group Gamma and let epsilon > 0. A set E subset of Gamma is a "weak epsilon-Kronecker set" if for every phi : E -> T there exists x in the dual of Gamma such that vertical bar phi(gamma) - gamma(x)vertical bar <= epsilon for all gamma is an element of E. When epsilon < root 2, every bounded function on E is known to be the restriction of a Fourier-Stieltjes transform of a discrete measure. (Such sets are called I-0.) We show that for every infinite set E there exists a weak 1-Kronecker subset F, of the same cardinality as E, provided there are not "too many" elements of order 2 in the subgroup generated by E. When there are "too many" elements of order 2, we show that there exists a subset F, of the same cardinality as E, on which every {-1, 1}-valued function can be interpolated exactly. Such sets are also I-0. In both cases, the set F also has the property that the only continuous character at which F . F-1 can cluster in the Bohr topology is 1. This improves upon previous results concerning the existence of I-0 subsets of a given E.