Examples in Helson sets

Robert Kaufman
1966 Bulletin of the American Mathematical Society  
A compact subset P of a locally compact abelian group G is said to be a Kronecker set in G [l, p. 97] if every continuous unimodular function on P is uniformly approximable on P by continuous characters of G. P is a Helson set [l, pp. 114-115] if for some €>0 and each n<EM(P): T being the dual of G. If P is a Kronecker set in G, P satisfies (H, 1) by [l, Lemma 5.5.1]. It was asked in [l] whether (H, 1) implies that P is a Kronecker set. Wik [2] constructed a class of counter-examples in the
more » ... examples in the real line; in this note a different type of construction is announced. Let X be a compact Hausdorff space and U the (abstract) group of continuous unimodular functions on X, V a subgroup of U which separates the points of X. Then X is embedded as a topological subspace of f and is a Kronecker set in f if and only if T is uniformly dense in U. We give below two examples in which T is a proper closed subgroup of U but for which (JÏ, 1) holds for measures in X. (a) X is the 1-torus and T the group of functions with winding number, or degree, zero. In this case the Kronecker condition holds on the complement of any arc, so (H, 1) holds. (b) X is the unit interval [0, 1 ] and T is the set of all functions e if , ƒ real and flfdx = 0. In this case U=T*C, C being the subgroup of constant functions. In (a) and (b) the groups T have the form exp {flr , where H is an additive subgroup of the real continuous functions on X. In each case H contains a dense subgroup H\ algebraically isomorphic to Z®Z@Z(& • • • ; the exponential mapping is an isomorphism onto T. In (a) Hi is the^ subgroup of trigonometric polynomials with coefficients in Z + V2Z; in (b) one uses the same coefficients with the generators {x n -l/(w + l):w^l}. Using the smaller subgroups of U determined by these subspaces we can embed X-»P W and have the same phenomenon in regard to measures in X. In view of Theorem 139
doi:10.1090/s0002-9904-1966-11454-4 fatcat:fgafq7noozdhlfu4cmtenz5pga