% LaTeX Article Template - using defaults \documentclass{article}[12pt] \usepackage{amssymb} \usepackage{latexsym} \pagestyle{myheadings} \markright{G. Hjorth} \usepackage{amssymb} \usepackage{latexsym} % Set left margin - The default is 1 inch, so the following % command sets a 1.25-inch left margin. %\setlength{\oddsidemargin}{-0.25in} % Set width of the text - What is left will be the right margin. % In this case, right margin is 8.5in - 1.25in - 6in = 1.25in. %\setlength{\textwidth}{7in} % Set top margin - The default is 1 inch, so the following % command sets a 0.75-inch top margin. %\setlength{\topmargin}{-0.25in} % Set height of the header %\setlength{\headheight}{0.5in} % Set vertical distance between the header and the text %\setlength{\headsep}{0.2in} % Set height of the text %\setlength{\textheight}{8.5in} % Set vertical distance between the text and the % bottom of footer %\setlength{\footskip}{0.4in} \pagestyle{myheadings} \markright{G. Hjorth} % Set the beginning of a LaTeX document \begin{document} \large \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newenvironment{proof}[1][Proof]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}]}{\hfill$\Box$\bigskip\end{trivlist}} \newenvironment{definition}[1][Definition]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}\bigskip} \newenvironment{example}[1][Example]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}\bigskip} \newenvironment{remark}[1][Remark]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}\bigskip} \newenvironment{notation}[1][Notation]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}\bigskip} \newenvironment{question}[1][Question]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}\bigskip} \newenvironment{examples}[1][Examples]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}\bigskip} \def\Ubf#1{{\baselineskip=0pt\vtop{\hbox{$#1$}\hbox{$\sim$}}}{}} \def\ubf#1{{\baselineskip=0pt\vtop{\hbox{$#1$}\hbox{$\scriptscriptstyle\sim$}}}{}} \def\Q{{\Bbb Q}} \def\Z{{\Bbb Z}} \def\N{{\Bbb N}} \def\n{{\cal N}} \def\m{{\cal M}} \def\T{{\Bbb T}} \def\R{{\Bbb R}} \def\C{{\cal C}} \def\G{{\cal G}} \def\l{{\cal L}} \def\p{{\mathcal P}} \def\a{{\cal A}} \def\b{{\cal B}} \def\B{{\cal B}} \def\e{\epsilon} \def\d{\delta} \title{On invariants for measure preserving transformations} % Enter your title between curly braces \author{G. Hjorth\footnote{The author gratefully acknowledges support from NSF grants DMS 99-70403 and DMS 96-22977.}} % Enter your name between curly braces %\date{\today} % Enter your date or \today between curly braces \maketitle \abstract{The classification problem for measure preserving transformations is strictly more complicated than that of graph isomorphism.\footnote{AMS Subject Classification: 04A15. Key words and phrases: Classification; measure preserving transformation; Polish group action.} \section{Preamble} \label{0} We consider the group $M_{\infty}$ of all invertible measure preserving transformations either on the unit interval or any other reasonable measure space. It seems natural to say that two of these transformations, $\sigma_1$, $\sigma_2$, are equivalent or {\it isomorphic} if there is a third, $\pi$, so that \[\pi\circ \sigma_1\circ\pi^{-1}=\sigma_2\:{\rm {a.e.}}\] To what extent can this equivalence relation be considered {\it classifiable}? In specific cases -- for instance $\sigma_1$, $\sigma_2$ both {\it Bernoulli} or {\it discrete spectrum} -- there are well accepted systems of complete invariants. However in the completely general context of arbitrary measure preserving transformations there is no known satisfactory system of complete invariants nor even a clear statement of what this would entail. For instance, Halmos in \cite{ha1} despairs of precisely formulating the problem but at page 1029 suggest that its solution should fulfill the ``the vague task of finding a complete set of invariants...'' At page 75 of \cite{ha2} he proposes that the central problem is to ``find usable necessary and sufficient conditions for the conjugacy of two measure preserving transformations.'' Some time later Weiss at page 670 of \cite{we} raises the problem of finding ``a set of invariants large enough so that if all invariants agree for two m.p.t. one can conclude that the m.p.t. are isomorphic.'' This article considers attempts to make this precise and ask abstractly whether the problem of classifiability could {\it in principle} have a positive solution. We present a clearly identifiable lower bound on the {\it classification difficulty} of the isomorphism relation for measure preserving transformations. One precise formulation of the problem would be to understand a {\it classifiable equivalence relation} on a Polish space to be one for which we may find a Borel assignment of reals or points in some other {\it standard Borel space} as complete invariants. This is the notion of classifiable suggested by the Glimm-Effros dichotomy of \cite{hakelo}. Indeed Feldman in \cite{feldman} takes exactly that position. Appealing to \cite{ornstein} he observes that the isomorphism relation for {\it Bernoulli} shifts allows real numbers to be assigned in a Borel manner as a complete invariant and uses \cite{ornsteinshields} to remark that such an assignment is already impossible for the class of measure preserving having the {\it property of K}. A more generous notion of classification, closer to the kinds we consider below, is already implicit in sources \cite{ha1}, \cite{ha2}, \cite{we}. In each case the results of \cite{hane} are accepted as providing a complete classification for the {\it discrete spectrum} measure preserving transformations. Here the invariants are not real numbers or single points in a standard Borel space, but rather countable sets of complex numbers. The significance of this is not in the use of complex numbers as against reals -- this is immaterial since all uncountable Polish spaces are Borel isomorphic. The significant feature of the invariants from \cite{hane} is that they have the form of a {\it countable unordered set} of points in a standard Borel space.\footnote{Superficially it might be thought that the ability to assign countable sets of points in a standard Borel space as a complete invariant is tantamount to classifiability in the sense of \cite{feldman}. In fact this is untrue -- there is in general no {\it canonical} way to encode or parameterize a countable set of complex numbers by a real, or by a point in any other separable completely metrizable (i.e. {\it Polish}) space. Indeed this can already been seen in the present context, as it is known that there is no Borel function that assigns to a discrete spectrum measure preserving transformation a complex number as a complete invariant (compare the start of $\S$5 below). More starkly, and just to allay suspicions that this may be a consequence of restricting to the Borel category, the techniques of \cite{solovay} are sufficient to demonstrate the consistency of set theory without the axiom of choice along with the non-existence of {\it any} function assigning complex numbers as complete invariants for discrete spectrum transformations. It is intrinsic to the methods of \cite{hane} that we obtain countable sets of points as complete invariants, and no amount of modification will squeeze it into the form requested by \cite{feldman}.} Thus we may in general ask for an equivalence relation $E$ on a standard Borel space $X$: \begin{question} {\bf Q1:} Does there exists a countable sequence $(f_i)_{i\in{\Bbb N}}$ of Borel functions, each $f_i:X\rightarrow {\Bbb C}$ such that for all $x_1, x_2\in X$ \[x_1 E x_2 \Leftrightarrow \{f_i(x_1): i\in{\Bbb N}\}=\{f_i(x_2): i\in{\Bbb N}\}?\footnote{Matt Foreman has shown that the procedure of \cite{hane} indeed obtains such a classification, that is to say, in the Borel category, for discrete spectrum mpt's.} \] \end{question} As suggested by Matt Foreman, we might hope that ``usable and sufficient conditions'' should at least make the relation of isomorphy Borel in $M_{\infty}\times M_{\infty}$. \begin{question} {\bf Q2:} Let Graph$(E)\subseteq X\times X$ be the set of $(x_1, x_2)$ for which $x_1 Ex_2$. Is Graph$(E)$ Borel in the product Borel structure on $X\times X$? (In future I will refer to this conclusion simply as the statement that $E$ be Borel.) \end{question} These are in general distinct notions. Given an equivalence relation $E$ on a Polish space $X$, classifiability in the sense of a single Borel function assigning points as complete invariants implies a positive answer to Q1. A positive answer to Q1 implies one for Q2. Neither of these implications reverse. Below we consider these questions for the specific case of $X=M_{\infty}$ and $E$ the equivalence relation of conjugacy, given by setting $\sigma_1 E\sigma_2$ if there exists $\pi\in M_{\infty}$ such that \[\sigma_1(x) =\pi\circ \sigma_2\circ \pi^{-1}(x) \: {\rm a.e.}\] We place on $M_\infty$ the customary Borel structure, described by \cite{feldman} and recalled in section \ref{1} below. \begin{theorem} \label{0.3} The conjugacy equivalence relation on $M_{\infty}$ is non-Borel. \end{theorem} In fact the situation is much worse than this alone would suggest. It turns out that the classification problem for measure preserving transformations encompasses the classification problem for arbitrary countable discrete structures -- countable groups, countable linear orderings, graphs, and so on. For ${\cal L}$ a countable language, Mod$({\cal L})$, the space of all ${\cal L}$-structures whose underlying set is ${\Bbb N}$, is naturally a standard Borel space; the details of this definition are recalled in section \ref{1} below, and discussed at length in many places, such as \cite{kechrisclassical}, \cite{hjorthkechris}, \cite{hjorth}. A specific example of such a collection is the space of (directed) graphs on $\N$, which by appeal to the corresponding characteristic function of the adjacency relation can be identified with $\{0,1\}^{\N\times\N}$ and given a natural topology. \begin{theorem} \label{0.4} If ${\cal L}$ is a countable language then there is a Borel function $\theta\! :{\rm Mod}({\cal L}) \rightarrow M_{\infty}$ such that for all $M, N\in$ Mod$({\cal L})$ \[M\cong N\Leftrightarrow \exists \pi\in M_{\infty}(\pi\circ \theta(M)\circ\pi^{-1}= \theta(N)\:{\rm {a.e.}}).\] \end{theorem} It is not known whether \ref{0.4} or \ref{0.3} can be obtained for the {\it ergodic} measure preserving transformations -- that is to say, whether we may have $\theta$ as in \ref{0.4} but with the further requirement that it always assume a value $\theta(M)\in M_{\infty}$ such that every Borel $\theta(M)$-invariant set is either null or co-null. These further and still open questions are of interest given the ergodic decomposition theorem, stating that every element of $M_{\infty}$ may be in some sense written as the integral of its ergodic components (see for instance \cite{ha2}, $\S$2.3 \cite{zimmer}, \cite{petersen}). In this context one could compare the conjugacy equivalence relation on the unitary representations of a discrete group: in the case of irreducible representations this is known to be not only Borel but actually $F_\sigma$ in an appropriate topology. (See \cite{effros}.) For the narrow case of ergodic transformations we only know the following: \begin{theorem} \label{0.6} There is a Polish group $G$ and a {``turbulent"} Polish $G$-space $X$ and a Borel function $\theta:X\rightarrow M_{\infty}$ such that \leftskip 0.5in \noindent (i) $\theta(x)$ is ergodic for all $x\in X$; \noindent (ii) $x_0E^X_G x_1$ if and only if \[\exists \pi \in M_\infty (\pi\circ \theta(x_0)\pi^{-1} = \theta(x_1)\] for all $x_0, x_1\in X$. \leftskip 0in \end{theorem} In other words, we may embed a {\it turbulent} orbit equivalence relation into the isomorphism relation on ergodic measure preserving transformations. In light of the results from \cite{hjorth}, this provides a succinct anti-classifiability result. In particular, the reduction in \ref{0.4} does not reverse: The classification problem for measure preserving transformations is {\it strictly} more complicated than for discrete countable structures. \begin{theorem} \label{0.7} For ${\cal L}$ a countable language, there is no Borel $\theta_1\! :M_{\infty}\rightarrow$ {\rm Mod}$({\cal L})$ such that for all $\sigma_1, \sigma_2\in M_{\infty}$ \[\exists \pi\in M_{\infty}(\pi\circ \sigma_1\circ\pi^{-1}= \sigma_2\:{\rm {a.e.}})\Leftrightarrow \theta_1(\sigma_1)\cong\theta_1(\sigma_2).\] \end{theorem} Indeed there is no such $\theta_1$ even with domain just the ergodic transformations. Actually we obtain in \ref{0.6}(i) that each $\theta(x)$ is in the class of {\it rank 2 generalized discrete spectrum} (see \cite{furstenberg}). This is an important detail: Since the discrete spectrum measure preserving transformations {\it do} admit classification by countable sets of complex numbers, and hence by countable models, we might have hoped for instance that the $\alpha$th level of the generalized discrete spectrum transformations admit complete invariants in something like the $\alpha$th iteration of the operation of taking all countable subsets applied to ${\Bbb C}$. It should not be thought that the results above are {\it fragile} to the choice of the Borel category. We can define more generous classes of reducibility and show that even with broader but still reasonable classes of functions -- of the kind that are encountered in Ulm invariants for abelian $p$-groups and the Scott analysis for countable structures -- there is no reduction of conjugacy on $M_{\infty}$ to isomorphism on countable structures, or equality on countable sets of reals, or indeed to any Borel equivalence relation. Finally I suppose it might be felt that the real problem is not that we are demanding Borel functions but more generally that we are requiring {\it any sort of definability} what so ever. In this way we might dream of some manner of classification, only without the invariants being produced in a ``effective'' manner. But not even that much can be hoped for. If $\approx$ is the conjugacy equivalence relation on $M_{\infty}$, $\cong$ isomorphism on countable structures, then by the techniques of \cite{solovay} it is consistent with ZF and enough of the axiom of choice to develop most classical mathematics that there be {\it no} injection: \[M_{\infty}/\!\!\approx\: \hookrightarrow {\rm {Mod}}({\cal L})/\!\cong.\] In particular if ${\cal P}_{\aleph_0}(A)$ denotes the collection of all countable subsets of a set ${A}$, then there will be no injection \[M_{\infty}/\!\!\approx\:\hookrightarrow {\cal P}_{\aleph_0}({\Bbb C}),\] nor \[M_{\infty}/\!\!\approx\: \hookrightarrow {\cal P}_{\aleph_0}({\cal P}_{\aleph_0}({\Bbb C})),\] and so on. Similarly it is consistent with ZF and a large fragment (DC) of choice that for any Borel equivalence relation on a Polish space $X$ there is no injection \[M_{\infty}/\!\!\approx\hookrightarrow X/E.\] In $\S$\ref{1} we give some definitions and present the outline of the proof for \ref{0.4}, which is in turn completed in sections \ref{2} and \ref{3}. Section \ref{4} embeds a turbulent orbit equivalence relation into the generalized discrete spectrum transformations. $\S$\ref{5} gives a proof of a known result to the effect that the natural equivalence relation on cocycles from the measure preserving action of a countable group into a compact group is Borel; this equivalence relation is closely related to the one needed in $\S$\ref{4}. It is also noted that the collection of measurable transformations conjugating a transformation $T$ to itself is compact if and only if $T$ has discrete spectrum. \bigskip In terms of background material needed for reading this paper, formally it does not assume much more than a general knowledge of elementary analysis, of the kind which would be found in an text such as \cite{zimmer}. However as a practical matter it would be more than helpful to have some acquaintance with ergodic theory. A knowledge of classical descriptive set theory in the sense of \cite{kechrisclassical} may also make the paper easier to read. Many of the results appeal to the modern theory of Borel equivalence relations; for this \cite{frst} and \cite{beckerkechris} are good references. The theory of turbulence is developed in \cite{hjorth}; the notation here largely follows the notation there. We indulge in all the usual sins. A measurable square summable function is identified with its equivalence class in $L^2$. We say ``everywhere" when we mean ``on all but a null set". \bigskip \noindent{\bf Acknowledgment:} I am grateful to Matthew Foreman for several illuminating discussions in the neighborhood of these topics and for pointing out the relevance of \cite{hane}. I am also very much indebted to the referee for an exceptionally thorough and penetrating report, and in particular for finding a serious mathematical error in the first draft. This first draft claimed that one can obtain \ref{0.4} by proving that isomorphism on countable torsion-free abelian groups is Borel complete in the sense of \cite{frst}; that proof of the Borel completeness of torsion-free abelian groups was erroneous. \newpage \section{Outline of the proof of \ref{0.4}} \label{1} The concept of ``Borel reducibility" is central to the arguments below. \begin{definition} Let $E$ and $F$ be equivalence relations on Polish spaces $X$ and $Y$. We say that $E$ is {\it Borel reducible} to $F$, written $E\leq_B F{\rm ,}$ if there is a Borel function \[\theta: X\rightarrow Y\] such that for all $x_1, x_2\in X$ we have \[x_1 E x_2\] if and only if \[\theta(x_1) F \theta(x_2).\] Naturally we write $E<_B F$ if $E\leq_B F$ holds but $F\leq_B E$ fails. \end{definition} This relation $\leq_B$ is clearly transitive and reflexive. \begin{examples} More detail, along with proofs of the various folklore assertions, can be found in \cite{hjorth}. (i) For $X$ a Polish space, id$(X)$ is the identity equivalence relation on $X$. If $E\leq_B$ id$(X)$ for any Polish space $X$ then we say that $E$ is {\it smooth}. (ii) $E_v$ is the Vitali equivalence relation on $\R$, given by the cosets of $\Q$. Here it is known that $E_v$ is not smooth. (iii) $E_0$ the equivalence relation of eventual agreement on infinite binary sequences. It is known that $E_0\leq_B E_v\leq_B E_0$. (iv) Let $2^{\N\times\N}$ be the space of functions from $\N\times\N$ to $\{0, 1\}$, with the topology of point wise convergence.\footnote{Here and elsewhere we identify 2 with $\{0,1\}$, and thus $2^{\N\times \N}$ is the space of all functions from $\N\times\N$ to $\{0,1\}$.} Following \cite{frst} we define $F_2$ by \[x_1 F_2 x_2\] if and only if \[\forall n\in \N \: \exists m_1, m_2\in \N (\forall k\in\N (x_1 (n, k)=x_2(m_2, k), x_1(m_1, k)=x_2(n, k))).\] Then ${\rm id}(\R)<_B E_0 <_B F_2.$ \end{examples} Two important classes of Polish spaces are those consisting of all measure preserving transformations of a Lebesgue space and those consisting of all ${\cal L}$-structures on $\N$ for some countable language ${\cal L}$. \begin{definition} Let $M_{\infty}$ be the group of Borel measure preserving bijections from $[0,1]$ to $[0,1]$ with identification of maps agreeing on a measure one set. For $(O_n)_{n\in{\Bbb N}}$ a basis of $[0,1]$ we obtain a separable metric $d_{\lambda}$ on $M_{\infty}$ by setting \[d_{\lambda}(\pi_1, \pi_2)=\sum 2^{-n} [\lambda(\pi_1(O_n)\Delta\pi_2(O_n))+\lambda(\pi_1^{-1}(O_n)\Delta\pi_2^{-1}(O_n))],\] where $\lambda$ is Lebesgue measure and $\Delta$ is used to denote symmetric difference: $X\Delta Y=(X\setminus Y)\cup (Y\setminus X)$. Let $\approx^*$ denote the conjugacy equivalence relation: \[\pi_1\approx^*\pi_2\Leftrightarrow \exists \sigma\in M_{\infty} (\sigma\circ \pi_1\circ \sigma^{-1}=\pi_2).\] \end{definition} \begin{definition} For ${\cal L}$ a countable language, let Mod$({\cal L})$ be the collection of all different ways we may place an ${\cal L}$-structure on the natural numbers $\N$. We then place a topology on this space by taking as subbasic open sets those of the form \[\{\m\in {\rm Mod}(\l)\mid (n_1,..., n_k)\in R^\m\},\] \[\{\m\in {\rm Mod}(\l)\mid (n_1,..., n_k)\not\in R^\m\},\] \[\{\m\in {\rm Mod}(\l)\mid f^\m(n_1,..., n_k)=m\},\] \[\{\m\in {\rm Mod}(\l)\mid f^\m(n_1,..., n_k)\not = m\},\] where $n_1,n_2,..., n_k, m$ range over finite sequences from $\N$, $R$ ranges over relation symbols in $\l$, and $f$ ranges over function symbols in $\l$. We let $\cong\! |_{{\rm Mod}(\l)}$ denote the isomorphism relation on these ${\cal L}$-structures. Thus $\m_1\cong\m_2$ if and only if there is a bijection $\sigma: \N\rightarrow\N$ with \[(n_1,..., n_k)\in R^{\m_1}\Leftrightarrow (\sigma(n_1) ,..., \sigma(n_k))\in R^{\m_2}\] and \[f^{\m_1}(n_1,..., n_k)=m\Leftrightarrow f^{\m_2}(\sigma(n_1),..., \sigma(n_k))=\sigma(m)\] all relation symbols $R$, function symbols $f$, and $n_1,..., n_k, m\in\N$. \end{definition} \begin{lemma} \label{5.1.2} $M_{\infty}$ is a Polish group; $d_\lambda$ is complete. \end{lemma} \begin{proof} This was shown in \cite{feldman}; a proof can also be found in chapter 2 of \cite{hjorth}. \end{proof} \begin{lemma} \label{5.1.2a} {\rm Mod}$(\l)$ is a Polish space whenever $\l$ is a countable language. \end{lemma} \begin{proof} This lemma should be obvious, since the space can be naturally identified with a suitable countable product of the Polish spaces $\N^{\{0, 1\}}$ and $\N^\N$. \end{proof} We wish to start working towards a proof that for any countable language $\l$ we have \[\cong\! |_{{\rm Mod}(\l)}\leq_B \approx^*.\] An equivalence relation $E$ on $X$ is said to be {\it Borel} if it is Borel as a subset of $X\times X$. It is then easily seen that the Borel equivalence relations are closed downwards under $\leq_B$. And thus since it is well known (see \cite{frst} or 6.16 \cite{hjorth}) that for many $\l$ one has $\cong\! |_{{\rm Mod}(\l)}$ non-Borel, a proof of \[\approx^*\leq_B \cong\! |_{{\rm Mod}(\l)}\] will in particular give that $\approx^*$ is non-Borel. It is rather cumbersome to be continually working with the full range of possible $\cong\! |_{{\rm Mod}(\l)}$ as $\l$ ranges over countable languages. Instead it will be convenient to work with a canonical example, which is already known to have maximal complexity in the $\leq_B$-ordering. For us a {\it graph} will be a directed graph where loops are possible but parallel edges are not. Thus we may naturally identify a graph on the underlying set $\N$ with a binary relation on $\N$, and this in turn may by consideration of the characteristic function be identified with $2^{\N\times \N}$. \begin{definition} Let Mod(Gph) be the Polish space $2^{\N\times\N}$ equipped with the product topology. We let the infinite symmetric group, $S_\infty$, consisting of all permutations of the natural numbers, act on $2^{\N\times\N}$ in the following manner: Given $\sigma\in S_\infty$ and $x\in 2^{\N\times\N}$, we define $\sigma\cdot x$ by \[(\sigma\cdot x)(n, m)=x(\sigma^{-1}(n), \sigma^{-1}(m)).\] We then let \[E^{\rm Mod(Gph)}_{S_\infty}\] denote the resulting orbit equivalence relation: \[x_1 E^{\rm Mod(Gph)}_{S_\infty} x_2 \] if and only if \[\exists \sigma\in S_\infty(\sigma\cdot x_1=x_2).\] \end{definition} Thus $E^{\rm Mod(Gph)}_{S_\infty}$ is the isomorphism relation for the space of all binary relations on $\N$. Of course as a space Mod(Gph) is nothing other than $2^{\N\times\N}$; it will be convenient to have this separate notation, to remind ourselves with Mod(Gph) that we are thinking of $2^{\N\times\N}$ as a Polish $S_\infty$-space in a specific way. \begin{lemma} \label{5.1.2b} If $\l$ is a countable language, then \[\cong\! |_{{\rm Mod}(\l)}\leq_B E^{\rm Mod(Gph)}_{S_\infty}.\] \end{lemma} \begin{proof} A proof of this well known folklore fact can be found in many places, including \cite{frst}. \end{proof} Thus the task of showing $\approx^*$ non-Borel has been reduced to showing that for any countable language $\l$ we have \[\cong\! |_{{\rm Mod}(\l)}\leq_B \approx^*,\] which has in turn been reduced to showing \[ E^{\rm Mod(Gph)}_{S_\infty}\leq_B \approx^*.\] In order to do this we will introduce one further equivalence relation, $E^{Y_2}_{G_\infty}$ defined shortly, and show in section \ref{2} first that \[E^{Y_2}_{G_\infty}\leq_B \approx^*,\] and then in section \ref{3} that \[E^{\rm Mod(Gph)}_{S_\infty} \leq_B E^{Y_2}_{G_\infty}.\] \begin{definition} Let $M(S_{\infty})= \{g\in(S_{\infty})^{[0,1]}: g$ Borel$\}$ be the group of measurable functions from $[0,1]$ to $S_{\infty}$, where $S_{\infty}$ is the infinite symmetric group on ${\Bbb N}$; we multiply pointwise, \[(g_1g_2)(x)= g_1(x)g_2(x){\rm ,}\] and identify functions that agree $\lambda$ a.e. \end{definition} Observe then that if $g^{-1}$ is the group theoretic inverse of $g\in M(S_{\infty})$ then $g^{-1}(x)=(g(x))^{-1}$ for ($\lambda$ a.e) $x\in[0,1]$. \begin{definition} Define $\psi: M_{\infty}\rightarrow$ Aut$(M(S_{\infty}))$ by the requirement that for $\pi\in M_{\infty}$, $x\in[0,1]$, $g\in M(S_{\infty})$ \[((\psi(\pi))(g))(x)=g(\pi^{-1}(x)).\] \end{definition} So $\psi(\pi)$ is the (group theoretic) automorphism of $M(S_{\infty})$ obtained by shift. In fact these groups all have natural topologies and $\psi$ is a homeomorphism from $M_\infty$ into the group of continuous automorphisms of $M(S_{\infty})$. \begin{definition} Form the semi-direct product $ M(S_{\infty}) \rtimes_\psi M_{\infty}$ in the usual way, so that for $g_0, g_1\in M(S_{\infty})$, $\pi_0, \pi_1\in M_{\infty}$ \[(g_0,\pi_0)(g_1, \pi_1)= (g_0(\psi(\pi_0))(g_1), \pi_0\pi_1).\] For short write $G_{\infty}=_{\rm df} M(S_{\infty}) \rtimes_\psi M_{\infty}$. \end{definition} \begin{definition} For $y\in 2^{\N\times \N}$ and $n\in\N$ we define $y(n, \cdot)\in 2^\N$ in the obvious way, by \[(y(n, \cdot))(m)=y(n, m).\] Let \[B_2=\{x\in 2^{{\Bbb N}\times{\Bbb N}}:n\not\! = m \Rightarrow x(n, \cdot)\not\! = x(m, \cdot)\}{\rm ;}\] $B_2$ is a $G_\delta$ subset of the Polish space $2^{\N\times\N}$ and hence Polish (see \cite{kechrisclassical} 3C). Let \[Y_2=\{y\in (B_2)^{[0,1]}\mid y \:{\rm Borel }\}\] be the space of measurable functions from $[0,1]$ to $B_2$ where we identify functions agreeing almost everywhere. We give this space the topology of convergence almost every where, so that \[f_n\rightarrow f\] if for almost every $x\in[0,1]$ we have $f_n(x)\rightarrow f(x)$. \end{definition} \begin{lemma} \label{5.1.7} $Y_2$ is a Polish space. \end{lemma} \begin{proof} Let $d'$ be a complete compatible metric on $B_2$. We obtain a complete metric $d_2$ by setting \[ d_2(y_0, y_1)=\int d'(y_0(x), y_1(x))\lambda(x)\] for $y_0, y_1\in Y_2$. \end{proof} \begin{definition} Let $G_{\infty}$ act on $Y_2$ as follows: Given $y\in Y_2$, \[y:[0,1]\rightarrow 2^{{\Bbb N}\times{\Bbb N}}{\rm ,}\] and \[(g, \pi)\in G_{\infty}{\rm ,} \] \[g:[0,1]\rightarrow S_{\infty}{\rm ,}\] \[\pi \in M_{\infty}{\rm ,}\] we define $(g, \pi)\cdot y:[0,1]\rightarrow 2^{{\Bbb N}\times{\Bbb N}}$ by \[(((g,\pi)(y))(x))(m, n)=(y(\pi^{-1}(x)))(g^{-1}(x)(m), n).\] (Here $g^{-1}$ is intended to be the group theoretic inverse of $g\in M(S_{\infty})$; thus $g^{-1}(x)=(g(x))^{-1}$.) Observing \[([(g_0,\pi_0)((g_1, \pi_1)(y))](x))(m,n)=[((g_1,\pi_1)(y))(\pi_0^{-1}(x))]((g_0^{-1}(x))(m), n)\] \[=(y(\pi^{-1}_1(\pi_0^{-1}(x))))(g^{-1}_1(\pi_0^{-1}(x))(g_0^{-1}(x))(m), n)\] \[=(y(\pi^{-1}_1(\pi_0^{-1}(x))))((((\psi(\pi_0))(g_1))^{-1}g_0^{-1})(x))(m), n).\] The last equality uses that the group operations for $M(S_{\infty})$ are calculated pointwise, and hence \[g_1^{-1}(\pi_0^{-1}(x))=(g_1(\pi_0^{-1}(x)))^{-1}\] and \[[(((\psi(\pi_0))(g_1))^{-1} g_0^{-1})](x)= [((\psi(\pi_0))(g_1^{-1}))(x)][g_0^{-1}(x)].\] %\[=y(\pi_1^{-1}(\pi_0^{-1}(x)))((\psi(\pi_0)(g_1)^{-1}))^{-1}g_0^{-1}(x))(m),n)\] Thus, \[[(((g_0,\pi_0)((g_1, \pi_1)(y)))(x))](m,n)= [y(\pi^{-1}_1(\pi_0^{-1}(x)))](((((\psi(\pi_0))(g_1))^{-1}g_0^{-1})(x))(m), n)\] \[=[((g_0(\psi(\pi_0)(g_1)),\pi_0\pi_1)(y))(x)](m,n),\] which establishes this to be an action. We then let $E^{Y_2}_{G_\infty}$ be the orbit equivalence relation on $Y_2$ resulting from this action. \end{definition} \begin{lemma} \label{5.1.7b} For $y_0, y_1\in Y_2$ we have \[y_0 E^{Y_2}_{G_\infty} y_1\] if and only if there is some $\pi\in M_{\infty}$ such that \[\lambda(\{x: \{(y_0(x))(n,\cdot): n\in{\Bbb N}\}= \{(y_1(\pi^{-1}(x)))(n,\cdot): n\in{\Bbb N}\}\})=1.\] \end{lemma} \begin{proof} The ``only if" part of the lemma is trivial. The ``if" direction uses the well known fact, a proof of which can be found in 18A \cite{kechrisclassical}, that any Borel set in the plane may be uniformized by a Lebesgue measurable function. \end{proof} \begin{notation} Following the Kuratowski-Mycielski theorem of 19A \cite{kechrisclassical}, choose $C\subset [0,1]$ to be a perfect set such that for any $k\in\N$ and \[x_1,x_2, ..., x_k\in C\] we have that if $x_i\not\! = x_j$ all $i0$ there is $g\in H_0$ of the form \[g=\sum_{j=1}^{j=k} c_j\chi_{A_j}\] with \[d_0(g, f_0)<\frac{\epsilon}{3},\] some $k\in\N$, $c_1,c_2,..., c_k\in{\Bbb C}$, of absolute value 1, and measurable subsets $A_1, ..., A_k$ of $\T$, each given by \[A_n=\{e^{2\pi i x}\mid a_n\leq x0$. Following the Kakutani-Rokhlin lemma (see page 48 \cite{petersen}) we may find $A\subseteq \T$ so that for some $n$ \leftskip 0.5in \noindent (i) $A$, $e^{2\pi i {\surd 2 } }A$, $e^{4\pi i {\surd 2 } }A$,... $e^{2n\pi i {\surd 2 } }A$, are pairwise disjoint\footnote{Here $e^{2\pi i {\surd 2 } }A=\{e^{2\pi i {\surd 2 } }\zeta:\zeta\in A\}$.}; \noindent (ii) $\lambda(\bigcup_{l1-\epsilon$. \leftskip 0in To obtain the existence of this set $A$ we apply Kakutani-Rokhlin to any $n>\frac{2}{\epsilon}$ to obtain $A$ so that \[\lambda(\bigcup_{l\leq n} e^{2l\pi i {\surd 2 } }A)>1-\frac{\epsilon}{2}\] and the sets \[A, e^{2\pi i {\surd 2 } }A, e^{4\pi i {\surd 2 } }A, ..., e^{2n\pi i {\surd 2 } }A\] are disjoint as in (i). Then we must have $\lambda(A)<\frac{\epsilon}{2}$ and (ii) follows as well. Now we may simply step around this sequence $A$, $e^{2\pi i {\surd 2 } }A$, $e^{4\pi i {\surd 2 } }A$,... $e^{2n\pi i {\surd 2 } }A$, defining $f|_{e^{2l\pi i {\surd 2 } }A}$ by induction on $l$ so that at each $\xi\in e^{2l\pi i {\surd 2 } }A$ we have $f(\xi) h_0(\xi) (f(\xi e^{2\pi i {\surd 2 } }))^{-1}=h_1(\xi)$. More formally, we let $f|_A$ just be constantly 1. Assuming inductively that $l1/4\lambda(A)\lambda(B).\] Since $h\mapsto T_h$ is continuous, the set of $h$ for which $T_h$ is ergodic is again $G_{\delta}$. However there is {\it some} $h$ for which $T_h$ is ergodic -- for instance $h:\T\rightarrow \T$, $\zeta\mapsto \zeta$ (see here \cite{befo} or \cite{furstenberg}). Thus by \ref{4.12}, the set of $h$ for which $T_h$ ergodic is a dense $G_{\delta}$. \end{proof} \begin{notation} Let $X_0=\{h\in X: T_h$ is ergodic $\}$. \end{notation} By \ref{4.14} we have that $X_0$ is $G$-invariant; since it is $G_\delta$ we have that it is a Polish $G$-space in its own right. For the convenience of the reader, we give the next definition only in the narrow context that is directly relevant. The more general definitions can be found in \cite{zimmer2} or \cite{petersen} $\S$2.4. \begin{definition} Let \[\rho:\T^2\rightarrow \T^2\] be an invertible measure preserving transformation of the form \[(\zeta, \xi) \mapsto (\zeta e^{2\pi i \surd{2}} , \hat{\rho}(\zeta, \xi))\] where $\hat{\rho}:\T^2\rightarrow \T$. Then a non-zero function $f\in L^2(\T^2)$ is said to be a {\it generalized eigenfunction} for $\rho$ if there is some $g\in L^2(\T)$, called a {\it generalized eigenvalue}, with the property that for all $(\zeta, \xi)\in \T^2$ we have \[f\circ \rho(\zeta, \xi)=g(\zeta)(f(\zeta, \xi));\] in other words, $f\circ \rho =\hat{g}f$ for $\hat{g}$ defined by \[\hat{g}(\zeta, \xi)=g(\zeta).\] \end{definition} The next couple of lemmas are standard; more general results, along with related facts, can be found in \cite{zimmer2}. \begin{lemma} \label{4.15a} Let $h\in X_0$ and let $f_1, f_2\in L^2(\T^2)$ be generalized eigenfunctions for $T_h$ with a common generalized eigenvalue. Then $f_1$ is a linear multiple of $f_2$. \end{lemma} \begin{proof} $f_1^{-1}f_2$ is invariant under $T_h$, and hence must be a constant function by ergodicity. \end{proof} \begin{lemma} \label{4.15b} Let $h\in X_0$. Then the only generalized eigenfunctions are \[(\zeta, \xi)\mapsto \xi^n k(\zeta)\] for some measurable function \[k: \T \rightarrow {\Bbb C}.\] \end{lemma} \begin{proof} Note that by Stone-Weierstrass, every function $f\in L^2(\T^2)$ can be written as \[f:(\zeta, \xi)\mapsto \sum_{n\in \Z}\xi^n k_n(\zeta)\] for some $k_n\in L^2(\T)$. Moreover decomposition is unique, since \[(\zeta, \xi)\mapsto \xi^nk_n(\zeta)\] and \[(\zeta, \xi)\mapsto \xi^m k_m(\zeta)\] are orthogonal for $n\not = m$. Now let us suppose that \[f:(\zeta, \xi)\mapsto \sum_{n\in \Z}\xi^n k_n(\zeta)\] has $g$ as its generalized eigenfunction. Then \[f\circ T_h :(\zeta, \xi)\mapsto \sum_{n\in \Z}\xi^n h(\zeta)^n k_n(\zeta e^{2\pi i \surd 2}).\] Then the uniqueness of the decomposition of $f\circ T_h$ gives for a.e. $\zeta\in\T$ that \[\xi^n h(\zeta)^n k_n(\zeta e^{2 \pi i \surd{2}})=g(\zeta) \xi^n k_n(\zeta).\] This means that any $k_n$ not identically zero gives rise to \[(\zeta, \xi)\mapsto \xi^n k_n(\zeta)\] as a function with generalized eigenfunction $g$; thus by \ref{4.15a} we have $k_n\equiv 0$ for all but a single $n$. \end{proof} \begin{lemma} \label{4.17} For all $h_0, h_1\in X_0$ \[T_{h_0}\approx T_{h_1}\Rightarrow h_0 E^X_G h_1.\] \end{lemma} \begin{proof} Fix $h_0, h_1$ with $T_{h_0}\approx T_{h_1}$, and let $\pi\in M_{\infty}$ witness this -- in that $\pi T_{h_0} \pi^{-1}=T_{h_1}$ $\lambda^2$ a.e. By ergodicity \[(\zeta, \xi)\mapsto \zeta\] is up to scalar multiples the only eigenfunction with eigenvalue $e^{2\pi i {\surd 2 } }$ for both $T_{h_0}$ and $T_{h_1}$. Thus \[\pi(\zeta, \xi)=(\zeta_0\zeta, \hat{\rho}(\zeta, \xi))\] for some fixed $\zeta_0\in\T$ and suitable $\hat{\rho}$. By replacing $h_0$ by $((\zeta_0)^{-1}, \bar{0}, 1)\cdot h_0$ we may assume that $\zeta_0=1$ and thus \[\pi(\zeta, \xi)=(\zeta, \hat{\rho}(\zeta, \xi)).\] By \ref{4.15b} we have that the only generalized eigenfunctions for $T_{h_i}$ ($i$ equal to either 0 or 1) are of the form \[(\zeta,\xi)\mapsto \xi^n k(\zeta)\] for some $n\in\Z$ and measurable $k:\T\rightarrow {\mathbb C}$. Thus we see that the generalized eigenfunctions of the form \[(\zeta,\xi)\mapsto \xi k(\zeta)\] \[(\zeta,\xi)\mapsto \xi^{-1} k(\zeta)\] have a privileged status, as the only generalized eigenfunctions which are able to generated the space $L^2(\T^2, \lambda^2)$ by the operations of multiplication, addition, and multiplication by linear combinations of the eigenfunctions $(\zeta, \xi)\mapsto \zeta^m$ some $m\in \Z$ (here by generate, I mean that they are dense in the sense of the Hilbert space norm on $L^2$). Note then that $(\zeta, \xi)\mapsto \xi$ must be sent to a generalized eigenfunction for $T_{h_1}$ of the form \[(\zeta,\xi)\mapsto \xi^{j} k(\zeta)\] where $j$ is either $1$ or $-1$. Thus we may assume \[\pi(\zeta, \xi)=(\zeta, \xi^{(-1)^i} k(\zeta))\] for some measurable $k:\T\rightarrow\T$, and so \[\pi^{-1}(\zeta, \xi)=(\zeta, \xi^{(-1)^i} (k(\zeta))^{(-1)^{i+1}}).\] Thus \[( e^{2\pi i {\surd 2 } }\zeta, h_0(\zeta)\xi)=\pi^{-1} T_{h_1} \pi(\zeta, \xi)=\pi^{-1} T_{h_1}(\zeta, k(\zeta)(\xi)^{(-1)^i})\] \[=\pi^{-1}(\zeta e^{2\pi i {\surd 2 }} , k(\zeta) h_1(\zeta)(\xi)^{(-1)^i}) =(\zeta e^{2\pi i {\surd 2 } }, (k(\zeta e^{2\pi i {\surd 2 } }))^{(-1)^{i+1}} (h_1(\zeta))^{(-1)^i}k(\zeta)^{(-1)^i}\xi).\] Thus for a.e. $(\zeta, \xi)$ we have \[ [(k(\zeta e^{2\pi i {\surd 2 } })^{-1}h_1(\zeta) k(\zeta)]^{(-1)^i} =h_0(\zeta),\] and so $h_0 E^X_G h_1$ as required. \end{proof} \begin{lemma} \label{4.18} Every orbit in $X$ is meager. \end{lemma} \begin{proof} Let \[A_0=\T\times \{e^{2 \pi i x}\mid \frac{1}{4}\leq x \leq \frac{3}{4}\}.\] \medskip \noindent{\bf Claim:} For any $O_1, O_2\subset X$ open, non-empty and $A$ measurable, there exist $h_1\in O_1$, $h_2\in O_2$, $k\in\N$, $k>0$. with \[\lambda^2(A\Delta T^k_{h_1}(A))<\frac{1}{13},\] \[\lambda^2(A_0\Delta T^k_{h_2}(A_0))>\frac{1}{4}.\] \noindent{\bf Proof of Claim:} Let $\hat{h}_1$ be the function on $\T$ \[\hat{h}_1: \zeta\mapsto 1,\] and let $\hat{h}_{\surd{3}}$ be the function \[\hat{h}_{\surd{3}}: \zeta\mapsto e^{2\pi i \surd{3}}.\] Then by \ref{4.12} we may find $(1, \bar{0}, f_1), (1, \bar{0}, f_2)\in G$ with \[h_1=_{\rm df} (1, \bar{0}, f_1)\cdot \hat{h}_1\in O_1,\] \[h_2=_{\rm df} (1, \bar{0}, f_2)\cdot \hat{h}_{\surd{3}}\in O_2.\] Since the continuous functions are dense in $L^1$ we may actually assume that $f_1$ and $f_2$ are continuous. Then \[h_1(e^{2\pi i x})=f_1(e^{2\pi i x})\hat{h}_1 (e^{2\pi i x})(f_1(e^{2\pi i (x+\surd{2})}))^{-1}\] \[=f_1(e^{2\pi i x})(f_1(e^{2\pi i (x+\surd{2})}))^{-1}\] and thus \[T_{h_1}: (e^{2\pi i x}, \xi)\mapsto (e^{2\pi i (x+\surd{2})}, f_1(e^{2\pi i x})(f_1(e^{2\pi i (x+\surd{2})}))^{-1}\xi)\] and \[(T_{h_1})^k: (e^{2\pi i x}, \xi)\mapsto (e^{2\pi i (x+k\surd{2})}, f_1(e^{2\pi i x})(f_1(e^{2\pi i (x+k\surd{2})}))^{-1}\xi).\] An exactly similar calculation gives \[(T_{h_2})^k: (e^{2\pi i x}, \xi)\mapsto (e^{2\pi i (x+k\surd{2})}, f_2(e^{2\pi i x}) e^{2k\pi i\surd{3}} (f_2(e^{2\pi i (x+k\surd{2})}))^{-1}\xi).\] The continuity of $f_1, f_2$ guarantees for each $\delta$ some corresponding $\hat{\delta}>0$ such that whenever \[|e^{2k\pi i \surd{2}} -1|<\hat{\delta}\] then for all $x\in [0, 1]$ \[|f_1(e^{2\pi i x})-f_1(e^{2\pi i (x+k\surd{2})})|, |f_2(e^{2\pi i x})-f_2(e^{2\pi i (x+k\surd{2})})|<\delta.\] Since $\surd{2}$ and $\surd{3}$ are rationally independent, we can therefore apply Kronecker's lemma (theorem 28 of \cite{morris}) and find $k$ with \[|f_1(e^{2\pi i x})-f_1(e^{2\pi i (x+k\surd{2})})|,\] \[|f_2(e^{2\pi i x})-f_2(e^{2\pi i (x+k\surd{2})})|\] both arbitrarily small for all $x\in [0,1]$ and $e^{2k\pi i \surd{3}}$ arbitrarily close to $e^{\pi i}$. Such $k$ clearly suffices. \hfill (Claim$\square$) \medskip Now let us choose a sequence of measurable sets $(B_i)_i$ such that for all $\pi\in M_\infty$ there is some $i\in\N$ with \[\lambda^2(\pi(A_0)\Delta B_i)<\frac{1}{13}.\] Let $V_i$ be \[\{\pi\in M_\infty\mid \lambda^2(\pi(A_0)\Delta B_i)<\frac{1}{13}\}.\] By \ref{4.14} suffices to show that for each $i$ the set \[\{(h_1, h_2)\in X\times X\mid \exists \pi \in V_i (\pi^{-1} \circ T_{h_1}\circ \pi =T_{h_2})\}\] is nowhere dense. But given any non-empty open $O_1, O_2\subset X\times X$ we can by the above claim find some non-empty open $U_1\subset O_1, U_2\subset O_2$ and $k\in\N$ such that for all $h_1\in U_1$, $h_2\in U_2$ \[\lambda^2(B_i\Delta T^k_{h_1}(B_i))<\frac{1}{13},\] \[\lambda^2(A_0\Delta T^k_{h_2}(A_0))>\frac{1}{4}.\] Fixing such $h_1, h_2$ and $\pi\in V_i$ we need to show \[\pi^{-1} \circ T_{h_1}\circ \pi\not = T_{h_2}.\] But since $\pi$ is measure preserving we have \[\lambda^2(A_0\Delta (\pi^{-1}T_{h_1}\pi)^k(A_0)) =\lambda^2(A_0\Delta \pi^{-1} (T^k_{h_1}(\pi (A_0))))\] \[=\lambda^2(\pi(A_0)\Delta T^k_{h_1}(\pi (A_0))),\] which by the triangle inequality is bounded by \[\lambda^2(\pi(A_0), B_i)+ \lambda^2(B_i, T^k_{h_1}(B_i)) +\lambda^2(T^k_{h_1}(B_i), T^k_{h_1}(\pi(A_0)))\] \[=\lambda^2(\pi(A_0), B_i)+ \lambda^2(B_i, T^k_{h_1}(B_i)) +\lambda^2(B_i, \pi(A_0))\] since $T_{h_1}^k$ is measure preserving, which in turn is bounded by \[\frac{1}{13}+\frac{1}{13}+\frac{1}{13}<\frac{1}{4}\] by assumption of $h_1\in U_1$ and $\pi\in V_i$. This is as required to show \[\lambda^2(A_0\Delta \pi^{-1} (T^k_{h_1}(\pi (A_0)))) \not = \lambda^2(A_0\Delta T^k_{h_2} (A_0)).\] \end{proof} \begin{definition} Let $H$ be a Polish group and $Y$ a Polish $H$-space. The action of $H$ on $Y$ is said to be {\it turbulent} if \leftskip 0.5in \noindent (i) every orbit is dense; \noindent (ii) every orbit is meager; \noindent (iii) for all $x, y\in Y$, $U\subseteq Y$, $V\subseteq H$ open with $x\in U$, $1\in V$, there exists $y_0\in [y]_H$ (the orbit of $y$) and such that for all open $U_0$ containing $y_0$ there is $k\in\N$, $(h_i)_{i< k}\subseteq V$, $(x_i)_{i\leq k}\subseteq U$ with \[x_0=x,\] \[x_{i+1}=h_i\cdot x_i,\] and \[x_k\in U_0.\] \leftskip 0in \end{definition} The usefulness of this concept is that it gives a sufficient condition for a degree of non-classifiability: As in \cite{hjorth} no turbulent action allows a Borel -- or even Baire measurable -- function reducing its orbit equivalence relation to isomorphism on countable structures. More generally, any equivalence relation into which we can embed a turbulent orbit equivalence relation will similarly be unclassifiable by countable structures considered up to isomorphism. \begin{lemma} \label{4.20} The action of $G$ on $X_0$ is turbulent. \end{lemma} \begin{proof} We already established that every orbit is dense and meager, so we are only left to show the ``local density" condition at (iii) from the definition of turbulence. For this purpose, fix $h_0, h_1\in X$ and $\epsilon>0$. It suffices to show there is some $n\in\N$ and there are some $g_0, g_1,...,g_{n-1}\in G$ such that \[h_{0,0}=h_0,\] \[h_{0,l+1}=g_l\cdot h_{0,l},\] we have \leftskip 0.5in \noindent (i) $d_G(g_l, 1_G)<\epsilon$ each $l3/\epsilon$. Appealing to Kakutani-Rokhlin we find $A\subseteq \T$ with \leftskip 0.5in \noindent (iv) $A$, $e^{2\pi i {\surd 2 } }A$, $ e^{4\pi i {\surd 2 } }A$,... $ e^{2n\pi i {\surd 2 } }A$ all disjoint; \noindent (v) $\therefore \lambda(A)<\epsilon/3$; \noindent (vi) $\lambda(\bigcup_{l\leq n} e^{2l\pi i {\surd 2 } }A) >1-\epsilon/3$. \leftskip 0in We now define $f_l:\T\rightarrow \T$ by induction on $l k, l