Rate of decay of concentration functions for spread out measures

Let G be a locally compact unimodular group and μ an adapted spread out probability measure on G. We relate the rate of decay of the concentration functions associated with μ. to the growth of a certain subgroup N _μ of G. In particular, we show that when μ. is strictly aperiodic (i.e., when N_μ= G) and G satisfies the growth condition V_G(m) ≥ Cm^D, then for any compact neighborhood K ⊂ G we have sup_g∈G μ^*n(gK) ≤ C′n ^-D/2. This extends recent results of Retzlaff [R2] on discrete groups for adapted probability measures.