Weakly almost periodic functions and thin sets in discrete groups

A subset E of an infinite discrete group G is called (i) an Rw-set if any bounded function on G supported by E is weakly almost periodic, (ii) a weak p-Sidon set (1 ≥ p ≥ 2) if on l¹(E) the l^p-norm is bounded by a constant times the maximal C^*-norm of l¹(G),(iii)a T-set if xE ∩ E and Ex ∩ E are finite whenever x ≠ e, and (iv) an FT-set if it is a finite union of T-sets. In this paper, we study relationships among these four classes of thin sets. We show, among other results, that (a) every infinite group G contains an Rw-set which is not an FT-set; (b) countable weak p-Sidon sets, 1 ≥ p< 4/3 are FT-sets.