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# Introduction

The purpose of this paper is to present some results and observations regarding the beta transformations introduced by Rényi [8]. Given , the beta transformation is defined for by . Parry [7] defined to be a beta number if the orbit is finite. If for some n, then is a simple beta number. If is a beta number which is not simple, then there is some smallest (the preperiod length) and (the period length) for which . For a simple beta number we define m = 0 and p to be the smallest integer with . Notice that m+p is the size of the orbit .

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

Thus, if is a beta number, then satisfies the polynomial equation , where

In particular, a beta number is an algebraic integer whose minimal polynomial divides . The degree of R is the beta degree of , which is in general larger than , the degree of , which is defined to be the degree of its minimal polynomial P. As we observed above, D is the size of the orbit .

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

We will refer to the sequence as the beta expansion (of 1) for . If this expansion is periodic, with preperiod length m and period length p, we denote it by , if m = 0, and by if m > 0. A simple criterion for to be the beta expansion for some was given by Parry [7],p.407: the sequence must dominate, in the sense of lexicographic order, the shifted sequences , for all . The number is then determined by (1.3). For periodic expansions, is also determined by (1.2).

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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