A fresh look at the notion of normality
Vitaly Bergelson, Tomasz Downarowicz and Michał Misiurewicz
Abstract
Let G be a countably infinite cancellative amenable semigroup
and let (Fn) be a (left)
Følner sequence in G. We introduce the notion of an
(Fn)-normal set in G and an
(Fn)-normal element of {0,1}G.
When G=(N,+) and Fn={1,2,...,n}, the
(Fn)-normality coincides with the classical notion.
We prove several results about (Fn)-normality, for
example:
- If (Fn) is a Følner sequence in G, such
that for every α∈(0,1) we have
∑nα-|Fn|<∞,
then almost every (in the sense of the uniform product measure
(½,½)G) x∈{0,1}G
is (Fn)-normal.
- For any Følner sequence (Fn) in G, there
exists an effectively defined Champernowne-like
(Fn)-normal set.
- There is a rather natural and sufficiently wide class of Følner
sequences (Fn) in (N,×), which we call
"nice", for which the Champernowne-like construction can be done
in an algorithmic way. Moreover, there exists a Champernowne-like
set which is (Fn)-normal for every nice Følner
sequence (Fn).
We also investigate and juxtapose combinatorial and Diophantine
properties of normal sets in semigroups (N,+)
and (N,×). Below is a sample of results that we obtain:
- Let A⊂N be a classical normal set. Then, for
any Følner sequence (Kn) in (N,×)
there exists a set E of (Kn)-density 1,
such that for any finite subset
{n1,n2,…,nk}⊂E,
the intersection
A/n1∩A/n2∩…∩A/nk
has positive upper density in (N,+). As a
consequence, A contains arbitrarily long geometric
progressions, and, more generally, arbitrarily long
"geo-arithmetic" configurations of the form
{a(b+ic)j, 0≤i,j≤k}.
- For any Følner sequence (Fn) in (N,+) there
exist uncountably many (Fn)-normal Liouville
numbers.
- For any nice Følner sequence (Fn) in
(N,×) there exist uncountably many
(Fn)-normal Liouville numbers.