For S a subordinator and Πn an independent Poisson process of intensity ne⁻ˣ,x>0 we are interested in the number Kn of gaps in the range of S that are hit by at least one point of Πn. Extending previous studies in [A.V. Gnedin, The Bernoulli sieve, Bernoulli 10 (2004) 79–96; A.V. Gnedin, J. Pitman, M. Yor, Asymptotic laws for compositions derived from transformed subordinators, Ann. Probab. 2006 (in press). http://arxiv.org/abs/math.PR/0403438, 2004; A.V. Gnedin, J. Pitman, M. Yor, Asymptotic laws for regenerative compositions: gamma subordinators and the like, Probab. Theory Related Fields (2006)] we focus on the case when the tail of the Lévy measure of S is slowly varying. We view Kn as the terminal value of a random process , and provide an asymptotic analysis of the fluctuations of , Kn, as n→∞, for a wide spectrum of situations.