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The purpose of this paper is to present some results and observations regarding the beta transformations introduced by Rényi [8]. Given , the

It is easy to see by induction (see below) that there are integers
with (the * digits* in the * beta expansion* of **1**) for
which , so that , where

The polynomial **R** is called the * characteristic polynomial* of . Parry
[7] showed that the roots of **R** other than lie in the disk . This was improved to by Solomyak [11]
and Flatto, Lagarias and Poonen [4] independently. In fact, Solomyak gives an
exact description of the compact subset of the plane which is the closure of the set of
all conjugates (other than ) of beta numbers.

Recall that a * Pisot number* (or * Pisot-Vijayaraghavan number*, or * PV
number*) is an algebraic integer for which all conjugates of
with satisfy . A * Salem number* is an
algebraic integer for which all conjugates satisfy
with at least one conjugate having . This implies that
is reciprocal, so is a conjugate of and all other conjugates
satisfy . Hence, is even and
[9], p.26.

Schmidt [10] showed that every Pisot number is a beta number. This was also proved
independently by Bertrand [1]. The idea behind the proof, which is based on the
box principle, goes back to a paper of Gelfond [5]. Schmidt also showed that if
every rational has a periodic expansion in base (i.e., the orbit
is finite), then must be a Pisot or Salem number. He conjectured
that the converse is true, so in particular that every Salem number would be a beta
number. A simple result from [3] is that a Salem number cannot be a *
simple* beta number: clearly the characteristic polynomial of a simple beta number
has a unique positive root, but a Salem number has two positive conjugates,
namely and .

In [3], we showed that if is a Salem number of degree 4, then is a
beta number. (We did not consider Schmidt's more general conjecture there; it remains
open for Salem numbers of all degrees). The explicit beta expansion of each such
was given. In all cases **m = 1**, i.e., (mod **1**) is a purely periodic
point of **T**. If has the minimal polynomial , then for fixed **a** the period is a unimodal function of **b** and takes
on values which lie in the set even .
Thus, for all Salem numbers of degree
**4**. No such bound seems to be true for Salem numbers of higher degree.

If we write , then the canonical beta expansion of **1** to base
is defined by the ``greedy'' algorithm: , and , for .
Clearly, for all **n**, and

Our main purpose here is to consider whether Salem numbers of degree 6 must be beta
numbers. We compute the beta expansions for all but **80** of the
Salem numbers of degree **6** and trace at most **15** (there are **11836** such numbers).
For all but 199 of these, we have . However, there is one example
for which , and two examples for which we can show
that but do not know whether . The
existence of such examples shows that the situation for Salem numbers of degree 6 is
certainly more complicated than the degree-4 case. However, the fact that there *
are* examples with **D** very large but finite perhaps supports Schmidt's
conjecture for Salem numbers of degree 6.

In § 6, we give a heuristic probabilistic argument based on ideas of the geometry of
numbers which makes plausible the observed difference between the cases and . The argument correctly predicts that will be
small relative to for Salem numbers of degree **4**. It predicts that almost all
Salem numbers of degree **6** will be beta numbers but that can be arbitrarily
large even for small . Moreover, it predicts that for each fixed even degree there should be a positive proportion of Salem numbers of degree **d** which
are * not* beta numbers, as well as a positive proportion that are beta numbers. We
present no computational evidence here for this latter prediction.
The heuristic argument suggests, for Salem numbers of fixed degree **d**, that the size of
is directly related to the size of . Here
denotes the discriminant of the polynomial **P**. We explore this
connection through some of the computed examples for **d = 6**.

The heuristic arguments
apply equally well to Schmidt's more general conjecture concerning the periodicity of
the beta expansions of rationals to a Salem number base, and predict that Schmidt's
conjecture is true for numbers of degree **4** or **6** and false for numbers of higher
degree. For **d = 4**, the model predicts that the size of the orbit for
fixed should be roughly proportional to . These predictions are not
tested here.

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