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Department of Mathematics, University of British Columbia,

Vancouver, B.C., Canada V6T 1Z2

boyd@math.ubc.ca

**Keywords** *Salem numbers, beta expansions, polynomials, computation *

**Subjclass** *Primary 11R06, 11K16; Secondary 11Y99 *

**
Abstract
**
For a given beta > 1, the * beta transformation* T = T_beta is defined for x
\in [0,1] by Tx := beta x (mod 1). The number beta is said to be a
* beta number* if the orbit \{T^n(1)\}_{n \ge 1} is finite, hence
eventually periodic. It is known that all Pisot numbers are beta numbers, and it is
conjectured that this is true for Salem numbers, but this is known only for Salem
numbers of degree 4. Here we consider some computational and heuristic evidence for
the conjecture in the case of Salem numbers of degree 6, by considering the set of
11836 such numbers of trace at most 15. Although the orbit is small for the
majority of these numbers, there are some examples for which the orbit size is shown to
exceed 10^9 and for which the possibility remains that the orbit is infinite. There
are also some very large orbits which * have* been shown to be finite: an example is
given for which the preperiod length is 39420662 and the period length is 93218808.
This is in contrast to Salem numbers of degree 4 where the orbit size is bounded by
2beta + 3. An heuristic probabilistic model is proposed which explains the difference
between the degree-4 and degree-6 cases. The model predicts that all Salem numbers
of degree 4 and 6 should be beta numbers but that degree-6 Salem numbers can have
orbits which are arbitrarily large relative to the size of \beta.
Furthermore, the model predicts that a positive proportion of Salem numbers of any fixed
degree \ge 8 will not be beta numbers. This latter prediction is not tested here.

- Introduction
- Recognizing Salem numbers of degree 6
- Are Salem numbers of degree 6 beta numbers?
- Patterns and regularities in the tables
- The computation of the beta expansion
- A probabilistic model
- References
- About this document ...

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