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Which Rationals are Ratios of Pisot Sequences?
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- Journal:
- Canadian Mathematical Bulletin / Volume 28 / Issue 3 / 01 September 1985
- Published online by Cambridge University Press:
- 20 November 2018, pp. 343-349
- Print publication:
- 01 September 1985
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- Article
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A Pisot sequence is a sequence of integers defined recursively by the formula -
. If 0 < a0 < a1 then an+1/an converges to a limit θ. We ask whether any rational p/q other than an integer can ever occur as such a limit. For p/q > q/2, the answer is no. However, if p/q < q/2 then the question is shown to be equivalent to a stopping time problem related to the notorious 3x + 1 problem and to a question of Mahler concerning the powers of 3/2. Although some interesting statistical properties of these stopping time problems can be established, we are forced to conclude that the question raised in the title of this paper is perhaps more intractable than it might appear.
. If 0 < a