Abstract. We obtain several results about sets of uniqueness for classes of Vilenkin systems which show that, in general, the more rapidly the parameters $p_k$ grow, the thinner the corresponding sets of uniqueness must be.

AMS Subject Classification (1991): 42C10; 43A75

Received February 3, 1999, and in revised form May 21, 1999. (Registered under 2706/2009.)