Introduction

If is not an integer, we can use T to set up an expansion for which the above description is still valid. Define and . Then , for . Clearly, for all n, and

You can verify this lack of uniqueness for yourself in case is the golden ratio. Notice that the relation enables one to replace a word 100 in such an expansion by 011. So clearly some numbers will have uncountably many expansions to this base, but only one of these is the beta expansion.

Recall the familiar argument which shows the periodicity of the b expansion of a rational number . It is clear that all the numbers are of the form with . Since there are only a finite number of possibilities for , there must be a repetition in the sequence and hence in the sequence . But determines uniquely, and hence is periodic and thus is periodic.

Let us consider whether there could be non-integers with the same property. It is easy to see by induction that , where

However, it does not seem likely at first glance that would be periodic if is a non-rational algebraic integer. It is clear that all of the numbers are in the ring , i.e. of the form

Recall that a Pisot number (or Pisot-Vijayaraghavan number, or PV number) is an algebraic integer for which all conjugates of with satisfy . (The conjugates are the roots of the minimal polynomial). For example, the golden ratio is a Pisot number since its conjugate satisfies . There are Pisot numbers of every degree, in fact there are such numbers in every number field.

Bertrand [1] and Schmidt [7] independently showed that if is rational and is a Pisot number then is periodic. It makes sense to test this somewhat unexpected result with a few experiments. For example, take and . This is easy to try numerically on a pocket calculator, or using Maple, and we find the values for for shown in Table 1. This certainly suggests that the expansion is purely periodic with period 8. We can verify this by carrying out the calculation in the ring , or let Maple do it and obtain the exact values for shown in Table 2.

Why is this periodic? The result becomes less mysterious if we realize that the numbers listed in Table 2 are the projections onto the real axis of points in the lattice , i.e. the set of integer combinations of and . Since is irrational, the set of such projections is dense in , so why does the above orbit only meet finitely many of these lattice points? This is where the condition that be a Pisot number is used. Notice that the conjugate of is . But since and all , we have

It is clear how this argument extends to any rational and any Pisot number . The only difference is that if has degree d, we are dealing with a lattice in . The bounds one obtains on the size of the orbit are quite large and do not suggest the many patterns one observes in practice. For a discussion of the expansions for Pisot numbers in , and an answer to the question of whether the complementary factor need be cyclotomic, see [3].

As mentioned above, Schmidt showed that if the -expansion of every rational is periodic then must be a Pisot or a Salem number. Recall here that a Salem number is an algebraic integer such that all conjugates satisfy with at least one case of equality. This implies that is reciprocal, so is a conjugate of and the minimal polynomial of is a reciprocal (palendromic) polynomial of even degree. Schmidt's conjecture is that the converse of this theorem is true, that is that if is a Salem number, then the -expansion of every rational is periodic.

The smallest degree for Salem numbers is 4. The behaviour of the preperiod and period lengths is quite tame in this case, as one can see from Table 3. , which gives the period and preperiod for each Salem number of degree 4 and trace 10. Each such number is identified by the middle coefficient of the minimal polynomial, i.e. satisfies :

The pattern observed in Table 3. is general, as we showed in [2], where we proved Schmidt's conjecture for and all Salem numbers of degree 4 (so is a beta number in Parry's terminology). The general conjecture remains open even for degree 4. The method used in [2] could likely be extended to other rationals with small denominators but it is unlikely to succeed for all rational . Explicitly, if , then 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, even for .

In contrast, for degree 6, we find a much more interesting behaviour, as one sees by inspecting the tables in [4]. Table 4 reproduces some of that data for the Salem numbers with minimal polynomials . In all cases but one, m = 1 and . But for c = -7, we have only been able to show that m + p > 1199978517, so the orbit in this case is very large, possibly infinite. Clearly c = -7 is notable not only for the size of its orbit but also the size of . Our discussion below will indicate why a small value of should be expected to lead to a large value of m + p. More precisely, we suggest a relation between the size of the orbit and the size of .

At first it is natural to suspect that in this case the orbit is infinite. In order to understand why we now feel this is not the case, we must consider the probabilistic model presented below and, in more detail, in [4]. What we believe is true here is that m and p are simply very large. Such large periods are possible. For example, for the Salem number with minimal polynomial , the preperiod and period of the orbit of 1 are m = 39420662 and p = 93218808. For other large values of m+p, see Table 2 of [4].

There are 11836 Salem numbers of degree 6 and trace at most 15. We have computed the complete expansion for all but 80 of these. For the most part there is a great deal of regularity in the distribution of , and the values of m and p are quite small: for all but 199 of the numbers surveyed, both m < 1000 and p < 1000. For 9609 of the numbers (81% of the total) we have m = 1, but, in contrast to the degree-4 situation, larger values of m do occur with a certain regularity. However, among the remaining 199 cases, there are at least 79 for which and at least two for which .

Our probabilistic model suggests that m + p will be large if is small relative to . A competing explanation might be that some peculiar arithmetic relation between the conjugates of on the unit circle is the cause of the large orbits. To see that this is not the case, consider the orbits of 1 when is a power of the smallest Salem number of degree 6, which has minimal polynomial . This data is given in Table 5. . The correlation between the large values of m+p and the large values of is evident.