In all that follows, G is an
infinite, nondiscrete, locally compact T0 abelian group with character group X and
Δ is a nonempty subset of X. In a standard proof of the existence of infinite (in fact,
perfect) Helson sets (see for example Hewitt and Ross) it is shown that each nonvoid
open subset of an arbitrary G contains a K-set (terminology of Hewitt and Ross)
homeomorphic to Cantor’s ternary set (or, in the terminology of Rudin, a Kronecker
set or a set of type Ka homeomorphic to the Cantor set). In this paper, it is shown
that K0,Δ-sets or Ka,Δ-sets homeomorphic to the Cantor set exist in profusion
in a large class of infinite nondiscrete locally compact T0 abelian groups
G, provided that Δ is not compact. (A nonvoid subset E of G is called a
K0,Δ-set if for every continuous function from E to T, the circle group,
and every 𝜖 > 0, there is a γ ∈ Δ such that |γ(x) − f(x)| < 𝜖 for all x ∈ E.
Let a be an integer greater than one. A nonvoid subset E of G is called a
Ka,Δ-set if it is totally disconnected and every continuous function on E
with values in the set of a-th roots of unity is the restriction to E of some
γ ∈ Δ.)
