On the fourier-stieltjes coefficients of cantor-type distributions

Abstract

For a givenn-tuple of non-negative numbers (p ₀,p ₁,...,p _n−1) whose sum is equal to unity let μ(t) denote the probability that Σ ^{= 1/∞} _j X _j /n ^j ≦t, where the independent random variablesX _j assume the values 0,1,...,n−1 with probabilitiesp ₀,p ₁,...,p _n−1 respectively. For mostn-tuples we obtain upper and lower bounds on |û(m)|; these estimates involve then-ary representation ofm, or in some cases of 2m, so that a very simple and explicit characterization of the sequences on whichû(m) approaches zero can be given. In particular, for the Cantor middle-third measure, corresponding to the triple (1/2, 0, 1/2), the following criterion is obtained.û(m) approaches zero on a sequenceT of integers if and only if Ω(2m) approaches infinity onT, where Ω(k) is the sum of the following three quantities associated with the ternary representation ofk: the number of runs of zeros, the number of runs of twos and the number of ones. The results obtained are easily extended to the case when then-tuple varies withj (subject to certain mild restrictions).