Limits in Compact Abelian Groups
by
J. Hart
University of Wisconsin Oshkosh
Coauthors: K. Kunen

For X a compact abelian group, and B an infinite subset of its dual G, let C_B be the set of all x in X such that the sequence < g(x) : g is in B > converges to 1. If F is a free filter on G, let D_F be the union, over all B from F, of the C_B. The sets C_B and D_F are subgroups of X. C_B always has Haar measure 0, while the measure of D_F depends on F. Whenever a subgroup H of X is equal to C_B for some countable subset B of characters on X, B is said to characterize H. We show that there is a filter F such that D_F has measure 0 but is not contained in any C_B, and hence is not characterized by any set B contained in G. This generalizes previous results for the special case where X is the circle group.

Paper reference: arXiv:math.GN/0408115

Date received: February 28, 2005