% analbase.tex 
% Analytic subspace of the reals without an analytic basis.
% Beros and Miller - Feb 2009 minor revisions June 2014

\documentclass[12pt]{article}

\usepackage{amsmath, amssymb, amsthm}

\def\rr{{\mathbb R}}
\def\qq{{\mathbb Q}}
\def\al{\alpha}
\def\si{\sigma}
\def\es{\emptyset}
\def\su{\subseteq}
\def\ga{\gamma}
\def\be{\beta}
\def\ep{\epsilon}
\def\om{\omega}
\def\span{\mathrm{span}}

\newtheorem{thm}{Theorem}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}

\renewcommand{\qedsymbol}{$\boxtimes$}

\begin{document}

\begin{flushright}
K. Beros\\
A. Miller\\
February 2009\\
\end{flushright}

\begin{center}
Analytic subspace of the reals without an analytic basis.
\end{center}

A Hamel basis is a basis for the reals $\rr$ considered as a
vector space over the field of rationals $\qq$.

\begin{thm}(Erdos and Sierpinski \cite{erdos})
There is no analytic Hamel base.
\end{thm}

\begin{proof}
Suppose on the contrary that $B$ is such a basis.  $\rr$ is the countable
union of sets of the form $q_1 B + \ldots q_n B$, where 
$q_1,\ldots,q_n \in\qq$.  
These sets are all analytic, hence have the property of Baire and
thus, by the Baire category theorem, for some $q_1, \ldots , q_n \in \qq$, 
the set $A = q_1 B + \ldots q_n B$ is non meager.  There is an interval $I
\su \rr$ such that $A$ is comeager in $I$.  Pick any distinct $x_1,
\ldots , x_n \in B$.  Let $q \in \qq $ and $J \su I$ be a
subinterval such that $q( x_1 + \ldots + x_n ) + A$ is comeager $J$.  Note that
$q$ can be chosen so that $q \neq q_i$, for each $i$.  This will ensure that
there will be none of the $x_i$ will be canceled out by elements of $A$.  Then

\[ W = (q( x_1 + \ldots + x_n ) + A) \cap A \neq \es \]

\noindent because both terms of the intersection above are comeager in $J$.
Any element of $W$ will be at the same time a linear combination of at least
$n+1$ elements of $B$ and also $n$ elements of $B$. This contradicts the linear
independence of $B$
\end{proof}

\bigskip

\begin{thm}
Every proper analytic subspace of $\rr$ is measure zero and meager.
\end{thm}

\begin{proof}

For the first claim, suppose that $A$ is an analytic subspace of $\rr$
has positive measure.  Then by Steinhaus' theorem, $A - A$ (the set of
differences of elements of $A$) contains a nontrivial interval.  Hence $A$ must
be all of $\rr$.


For the second claim, suppose that $A$ is an analytic subspace which is
non-meager.  $A$ has the property of Baire and hence there is an open interval
$I$ in which $A$ is comeager.  Fix any $\al \in \rr$.  Let $q \in
\qq $ be small enough that $q\al$ is less than the length of $I$.  Then
$J = (q\al + I) \cap I$ is non empty and $q\al + A$ and $A$ are both
comeager in $J$.  In other words, there exist $x, y \in A$ such that 
$q\al+x = y$. Hence $\al = \frac{1}{q}(y - x) \in A$.  We see that $A = \rr$.

\end{proof}

\bigskip

The following answers a question raised by Ashutosh Kumar.

\begin{thm}
There exists a proper (and hence meager) analytic subspace of $\rr$ with
no analytic basis.
\end{thm}

\bigskip

We begin by describing the subspace in question.
\par Let $\ep_n$ be a decreasing sequence of positive rational numbers
such that for every $k$ and each $N \geq k$,

$$\sum_{n > N} k\ep_n \leq \frac{1}{4} \ep_N$$

This condition requires the $\ep_n$ to be a very rapidly
decreasing sequence.  Now let $P$ be the set defined by

$$P = \{ \sum_{n \in \om} x_n \ep_n : x_n \in \{-1 , 0 , 1\}\}$$



$P$ is essentially a very sparse Cantor set.  We now take the subspace $A$ to be
$\span(P)$. Our first objective is to show that $A$ is a proper
subspace of $\rr$.  To this end, we make the following observations: $A$
is the union of all sets of the form $q_1 P + \ldots + q_nP$, where the $q_i$
are rational numbers. By taking common denominators, we can write such sets as
$\frac{1}{m}(p_1 P + \ldots + p_n P)$, for some $p_1, \ldots , p_n \in \om$.
If we let $k = p_1 + \ldots + p_n$, then

$$(p_1 P + \ldots + p_n P) \subset \underbrace{P + P + \ldots + P}_{k\mbox{
times}}$$

We give this latter set the name $Q_k$.  Observe that $Q_k$ can be described by

$$Q_k = \{ \sum_{n \in \om} x_n \ep_n : x_n \in \{-k, -k+1, \ldots , k-1
, k\}\}$$

$p_1 P + \ldots + p_n P \subset Q_k$ and hence $\frac{1}{m}(p_1 P + \ldots +
p_n P) \subset \frac{1}{m}Q_k$.  Note that of course each $\frac{1}{m}Q_k$ is
also a subset of $A$.

\par Before proceeding, note that throughout we will use the notation $\hat
\si$ for the rational number $\sum_{n < |\si|} \si (n) \ep_n$,
where $\si$ is a finite sequence of integers.

\par We now show that $A$ is a proper subspace via the following two lemmas.

\bigskip

\begin{lem}
Suppose that $\si, \tau \in \{-k, \ldots , k\}^{< \om}$ such that
$|\si| = |\tau| > k$.  If $\si <_{\mbox{lex}} \tau$, then every point in
$Q_k^\si$ is less than every point in $Q_k^\tau$.
\end{lem}

\begin{proof}  It suffices to prove this lemma for the case in which there
exists $\ga$ such that $\si = \ga \hat \ i$ and $\tau = \ga  \hat \ 
(i + 1)$, for some $i \in \{-k, \ldots , k-1\}$.  Let $M = |\ga|$.  The
greatest element of $Q_k^\si$ is 

$$\al = \hat \si + \sum_{n > M} k \ep_n = \hat \ga + i \ep_M
+ \sum_{n > M} k \ep_n$$

and the least element of $Q_k^\tau$ is 

$$\be = \hat \tau - \sum_{n > M} k \ep_n = \hat \ga + (i \ep_M -
\sum_{n > M} k \ep_n)$$

Therefore,  $$\be - \al  = \ep_M - \sum_{n>M} 2k \ep_n \geq 
\frac{1}{2} \ep_M >0$$

\end{proof}

\bigskip

\begin{lem}
Each $Q_k^\si$ is nowhere dense, for $|\si| > k$.
\end{lem}

\begin{proof}

Fix any interval $I$ such that $I \cap Q_k^\si \neq \es$.  Choose $\tau
\supseteq \si$ such that $\hat{\tau\hat \ i}, \hat{\tau \hat \ (i + 1)} \in
I$, for some $i \in \{-k , \ldots , k-1\}$.  Then every element of
$Q_k^{\tau\hat \ i}$ is less than every element of $Q_k^{\tau\hat \ (i+1)}$ by
the previous lemma.  Therefore, between all $Q_k^\si$ are closed sets, we may
take an interval $J$ between $Q_k^{\tau\hat \ i}$ and $Q_k^{\tau\hat \ (i+1)}$. 
$J$ is disjoint from $Q_k^\si$, because $Q_k^\si$ is the disjoint union of
$Q_k^\ga$ for $|\ga| = |\tau| + 1$ and by the previous lemma, no such
$Q_k^\ga$ intersects $J$.

\end{proof}

\bigskip

This shows that each $Q_k^\si$ is nowhere dense.  Hence $Q_k$ is as well,
being a finite union of such $Q_k^\si$.  It follows that each $\frac{1}{m}
Q_k$ is nowhere dense and hence $A = \bigcup_{m,k \in \om} \frac{1}{m}Q_k$ 
is meager.  $A$ is therefore proper.

\par Now we get to our main claim.

\bigskip

\begin{lem}

$A$ has no analytic basis as a vector space over $\qq $.

\end{lem}

We begin with some remarks about the set $P$.  As in the above lemma, for
$\si \in \{-1, 0, 1\}^{< \om}$, we define

\[ N_\si = \{ \sum_{n \in \om} x_n \ep_n : x_n \in \{ -1, 0 , 1\} \
\& \ \si (n) = x_n \mbox{ for } n < |\si| \}\]

Note that although the $N_\si$ are closed sets in $\rr$ (and hence in
$P$), they are also relatively open in $P$.  In fact, they form a base for
the relative topology on $P$.

\begin{proof}[Proof of Lemma 6]

Suppose towards a contradiction that $B$ is an analytic basis for $A$.  We may
assume, without loss of generality, that $1 \in B$.  Otherwise, suppose that
$x_1, \ldots , x_n \in B$ and $q_1, \ldots q_n \in \qq $ are such that $1 =
q_1 x_1 + \ldots + q_n x_n$.  Then $[(q_1 - 1)x_1 + q_2 x_2 + \ldots q_n x_n] +
x_1 = 1$.  Hence $[(q_1 - 1)x_1 + q_2 x_2 + \ldots q_n x_n] + B$ is an analytic
basis for $A$ which contains 1.

\par Since $B$ is a basis, the generating set $P$ of $A$ must be covered by a
union of set of the form

$$q_1 (B \cap I_1) + \ldots + q_n (B \cap I_n)$$

\noindent where $q_1, \ldots , q_n \in \qq $ and $I_1, \ldots , I_n$ are
pairwise disjoint intervals with rational endpoints.  To avoid confusion later
on, we assume here that all $q_j$ are nonzero and that each $B \cap I_j$ is
nonempty.

$P$ is an uncountable closed set and hence a Baire space when regarded as a
topological subspace of $\rr$.  Because the union described above is
countable, the Baire category theorem yields that there are 
$q_1, \ldots , q_n , I_1 , \ldots , I_n$ 
as above such that

$$W = q_1 (B \cap I_1) + \ldots + q_n (B \cap I_n)$$

\noindent is non-meager in $P$.  $W$ is analytic, hence has the Baire property.
We therefore obtain $\si \in \{-1, 0 , 1\}^{ \om}$ such that $W$ is
comeager in $N_\si$.  (Because the $N_\si$ are a base for the relative
topology on $P$.)

We now define a homeomorphism $\pi$ of $N_\si$ as follows: If $z \in
N_\si$, then $z = \hat \si + \sum_{n \geq |\si|} x_n \ep_n$, for
some sequence $\langle x_n : n \in \om \rangle \in \{-1,0,1\}^\om$.  We
define $\pi (z) = \hat \si - \sum_{n \geq |\si|} x_n \ep_n$.  It is
clear that $\pi$ is an autohomeomorphism of $N_\si$.

It follows that $\pi^{-1} (W)$ is also comeager in $N_\si$ and hence $W
\cap \pi^{-1} (W) \neq \es$.  Let $z \in W \cap \pi^{-1} (W)$. Then $z ,
\pi(z) \in W$.  Note that we may assume that $z$ (and hence $\pi (z)$) are
irrational.  This follows from the fact that the rationals are meager in
$N_\si$.

We may now take $x_j, y_j \in B \cap I_j$ such that

\[ z = q_1 x_1 + \ldots + q_n x_n \]
\[ \pi (z) = q_1 y_1 + \ldots + q_n y_n \]

Thus

\[ z + \pi (z) = q_1 (x_1 + y_1) + \ldots + q_n (x_n + y_n) \]

By the definition of $\pi$, $z + \pi (z) = \hat \si \in \qq $.  Note
that since the $I_j$ are disjoint, for each $j$ and $i \neq j$ $x_j \neq x_i,
y_i$. Further, because $z \notin \qq $, $z \neq \pi(z)$, we have that for
at least one $j$, $x_j \neq y_j$.  We have therefore expressed a rational number
(namely $\hat \si$) as a sum of $n + 1$ distinct elements of the basis $B$.
On the other hand, $1 \in B$ and any rational can be expressed as a rational
scalar multiple of 1, i.e. a linear combination of length 1.  By independence,
such linear combinations are unique and so the above leads to a contradiction.


\end{proof}


\bigskip

We conclude with some further notes.

\bigskip

\begin{thm}\label{seven}
For all $\al > 2$ there exists a $\qq$--subspace $A$ of $\rr$
which is $\mathbf \Sigma_\al^0$, but not $\mathbf \Pi_\al^0$.
\end{thm}

\begin{proof}

Let $C \su \rr$ be a perfect, linearly independent set.  Choose $B
\su C$ which is $\mathbf \Sigma_\al^0$, but not $\mathbf
\Pi_\al^0$.  Take $A$ to be the linear span of $B$.

First of all, $A$ is not $\mathbf \Pi_\al^0$.  To see this, observe that, by
the independence of $C$, $A\cap C = B$.  If $A$ were $\mathbf \Pi_\al^0$,
then $B$ would be as well.

Secondly, $A$ is $\mathbf \Sigma_\al^0$.  Observe that $A$ is the union of
sets of the form.

\[ q_1(B \cap I_1) + \ldots + q_n(B \cap I_n) \]

Where the $q_i$ are nonzero rational numbers and the $I_i$ are disjoint
intervals with rational endpoints.  We can define a homeomorphism

\[ \prod_{i = 1}^n (C \cap I_i) \rightarrow q_i (C \cap I_i) + \ldots + (C \cap
I_n) \]

by $\langle x_1, \ldots , x_n \rangle \mapsto q_1x_1 + \ldots + q_nx_n$.  Under
this map, $\prod_{i = 1}^n (B \cap I_i)$ maps onto $q_1(B \cap I_1) + \ldots +
q_n(B \cap I_n)$.  Hence this latter set of is of the same Borel class as $B$,
namely $\mathbf \Sigma_\al^0$.  Since the union above is countable, $A$ is
also $\mathbf \Sigma_\al^0$.

\end{proof}

Theorem \ref{seven} is also a consequence of Theorem 2.5 of
Farah and Solecki \cite{farah} but has a shorter proof.

\begin{thm}
For all $\alpha > 3$, there exists a $\mathbb Q$-subspace $W$ of $\mathbb R$ which is $\mathbf \Pi^0_\alpha$ and not $\mathbf \Sigma^0_\alpha$.
\end{thm}

\begin{proof} Let $C \subset \mathbb R$ be a perfect, independent set over
$\mathbb Q$. Let $A_0 \supset A_1 \supset \ldots$ be subsets of $C$ which are
$\mathbf \Sigma^0_{<\alpha}$ and such that $A =  \bigcap_{n \in \omega} A_n$ is
$\mathbf \Pi^0_\alpha \setminus \mathbf \Sigma^0_\alpha$.  Let $W_n =
\mathrm{span}_{\mathbb Q}(A_n)$ and $W =  \bigcap_{n \in \omega} W_n$.  Then
each $W_n$ is $\mathbf \Sigma^0_{<\alpha}$ as in the proof of 
Theorem \ref{seven}.  Thus $W$ is $\mathbf \Pi^0_\alpha$, but not $\mathbf \Sigma^0_\alpha$. 
If $W$ were $\mathbf \Sigma^0_\alpha$, then $A = W \cap C$ would be as well.
\end{proof}


\begin{thebibliography}{99}

\bibitem{erdos}
Erd\"{o}s, P.; Some remarks on set theory. Proc. Amer. Math. Soc. 1, (1950).
127-141. 

\bibitem{farah} Farah, Ilijas; Solecki, S{\l}awomir;
Borel subgroups of Polish
groups. Adv. Math. 199 (2006), no. 2, 499-541.

\end{thebibliography}

\end{document}
