247 research outputs found
On the asymptotic behaviour of the correlation measure of sum-of-digits function in base 2
Let denote the number of digits "" in a binary expansion of any
. We study the mean distribution of the quantity
for a fixed positive integer .It is shown that solutions
of the equationare uniquely identified by a finite
set of prefixes in , and that the probability distribution of
differences is given by an infinite product of matrices whose coefficients
are operators of .Then, denoting by the number of
patterns "" in the binary expansion of , we give the asymptotic
behaviour of this probability distribution as goes to infinity as well
as estimates of the variance of the probability measure $\mu\_a
A new class of Ornstein transformations with singular spectrum
It is shown that for any family of probability measures in Ornstein type
constructions the corresponding transformation has almost surely a singular
spectrum. This is a new generalization of Bourgain's theorem, the same result
is proved for Rudolph's construction.Comment: 200
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