ε-Kronecker and I0 sets in abelian groups, I:: arithmetic properties of ε-Kronecker sets

A subset E of the locally compact abelian group Gamma is "epsilon-Kronecker" if every continuous function from E to the unit circle, can be uniformally approximated oil E by a character with error less than epsilon. The set E subset of Gamma is I-0 if every bounded function on E can be interpolated by the Fourier-Stieltjes transform of a discrete measure on the dual group. We show that if epsilon < root 2 then an E-Kronecker set is I-0, but this is not true for at least one root 2-Kronecker set epsilon-Kronecker sets in Z need not be finite unions of Hadamard sets. As with Sidon sets, epsilon-Kroriecker sets with epsilon < 2 do not contain arbitrarily long arithmetic progressions or large squares. When epsilon < root 2 they can contain only a bounded number of pairs with common differences and their step length tends to infinity. Related results and examples are given to show the sharpness of these results.