Ehsaan Hossain, Department of Pure Mathematics, University of Waterloo
"Return Sets on Algebraic Varieties"
Let
$\varphi:X\rightarrow
X$
be
an
algebraic
mapping
of
an
algebraic
variety
$X$,
so
that
each
point
$x\in
X$
has
an
orbit
$\{x,\varphi(x),\varphi^2(x),\ldots\}$.
We
consider
the
set
of
``times''
when
the
orbit
of
$x$
returns
to
a
given
closed
set
$C$:
$$E=E_\varphi(x,C)
=
\{n\in
\mathbb{N}
:
\varphi^n(x)\in
C\}.$$
This
is
called
an
\textbf{algebraic
return
set}.
Our
main
question
is:
which
subsets
of
$\mathbb{N}$
can
appear
as
algebraic
return
sets?
It
is
conjectured
that
$E$
can
only
be
infinite
if
it
contains
an
infinite
arithmetic
progression,
which
means
that
$x$
must
return
to
$C$
periodically.
In
2014,
Bell--Ghioca--Tucker
proved
this
periodicity
when
$E$
has
\textit{positive
density}.
Here
the
\textbf{density}
of
a
set
$S\subseteq
\mathbb{N}$
is
defined
to
be
the
limiting
proportion
of
members
of
$S$
amongst
the
first
$n$
integers:
$$\delta(S)
=
\limsup_{n\rightarrow\infty}\frac{|S\cap
\{1,\ldots,n\}|}{n}.$$
But
their
theorem
does
not
rule
out
infinite
zero-density
sets
such
as
$T=\{
n\in
\mathbb{N}
:
\text{the
$3$-ary
expansion
of
$n$
contains
no
$2$'s}\}$,
which
is
an
``automatic''
set
containing
no
arithmetic
progression
---
can
this
$T$
be
a
return
set?
We
will
show
that
it
is
not
by
adapting
BGT's
proof,
realizing
their
argument
as
a
disguised
version
of
the
Poincar\'{e}
Recurrence
Theorem
from
ergodic
theory.
MC 5403