Vol. 44, No. 1, 1973

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Existence of special K-sets in certain locally compact abelian groups

Frank Belsley Miles

Vol. 44 (1973), No. 1, 219–232

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.)

Mathematical Subject Classification 2000
Primary: 43A46
Received: 25 August 1971
Revised: 25 April 1972
Published: 1 January 1973
Frank Belsley Miles