# Normal sets in amenable semigroups

## Vitaly Bergelson, Tomasz Downarowicz and Michal Misiurewicz

### Abstract

Let (*F*_{n}) be a (left) Folner sequence in a countable
amenable semigroup *G* which is embeddable in a group. We
introduce the notion of (*F*_{n})-normal set in *G*
and (*F*_{n})-normal sequence in
{0,1}^{G}. When *G* = (**N**,+)
and *F*_{n} = {1,2,...,n}, the
(*F*_{n})-normality coincides with the classical notion.
We prove several results about this notion, for example:
- If (
*F*_{n}) is a Folner sequence in a countable
amenable group *G*, such that
∑_{n }e^{-|Fn|} < ∞ then
almost every *x*∈{0,1}^{G} is
(*F*_{n})-normal.
- In any amenable semigroup which is embeddable in a group, given
any Folner sequence (
*F*_{n}), there exists a
Champernowne-like effective (*F*_{n})-normal set.

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
Folner sequence (*K*_{n}) in (**N**,×) there exists a set *E*
of (*K*_{n})-density 1, such that for any finite subset
{n_1,n_2,...,n_k} ⊂ *E*, the
intersection *A*/*n*_{1} ∩
*A*/*n*_{2} ∩ ...
∩ *A*/*n*_{k} 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}.
- There is a rather natural and sufficiently large class of Folner
sequences (
*F*_{n}) in (**N**,×), which we call
"nice", and which have the property that there exists a Liouville
number, whose binary expansion is (*F*_{n})-normal.
- In (
**N**,×) there exists a Champernowne-like set which
is (*F*_{n})-normal for every nice Folner sequence
(*F*_{n}).
- Any set in (
**N**,×) which is normal with respect to
every nice Folner sequence contains, for each *k*
∈ **N**, solutions *a,b,c* of the
equation *ab*=*c*^{k}.