1) Every infinite, Abelian compact (Hausdorff) group K admits 2 |K|many dense, non-Haar-measurable subgroups of cardinality |K|. When

For a recursively defined sequence u := (un) of integers, we describe the subgroup tu(T) of the elements x of the circle group T satisfying limn unx = 0. More attention is dedicated to the sequences satisfying a secondorder recurrence relation. In this case, we show that the size and the free-rank of tu(T) is determined by the asymptotic behaviour of the ratios qn = un u n−1 and we extend previous results of G. Larcher, C. Kraaikamp, and P. Liardet obtained from continued fraction expansion.

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We prove an algebraic and a topological decomposition theorem for complete D-lattices (i.e., lattice-ordered effect algebras). As a consequence, we obtain a HammerSobczyk type decomposition theorem for modular measures on D-lattices.

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