Pisot sequence
The standard Pisot -sequence E ( a _ { 0 } , a _ { 1 } ) is the sequence of positive integers \{ a _ { n } \} defined for 0 < a _ { 0 } < a _ { 1 } by the recursion
\begin{equation*} a _ { n } = N \left( \frac { a _ { n - 1} ^ { 2 } } { a _ { n - 2} } \right), \end{equation*}
if n \geq 2, where N ( x ) = \lfloor x + 1 / 2 \rfloor denotes the nearest integer function. For example E ( 3,5 ) = \{ 3,5,8,13 , \dots \}. If a _ { 1 } > a _ { 0 } + 2 \sqrt { a _ { 0 } }, one can show that a _ { n } = \lambda \theta ^ { n } + \epsilon _ { n }, where \lambda > 0 and \theta = \theta ( a _ { 0 } , a _ { 1 } ) > 1 and where
\begin{equation*} \operatorname {lim} \operatorname{sup} | \epsilon _ { n } | \leq \frac { 1 } { 2 ( \theta - 1 ) ^ { 2 } }. \end{equation*}
Thus, at least when \theta > 2, it is clear that \lambda \theta ^ { n } is badly distributed modulo 1 (cf. also Distribution modulo one). These sequences were originally considered in [a7] for this reason.
The \theta ( a _ { 0 } , a _ { 1 } ) are called E-numbers. The set E of E-numbers is dense in the interval [ 1 , \infty ). E contains the set H of those \theta for which there is a \lambda > 0 such that \| \lambda \theta ^ { n } \| \rightarrow 0. (Here \| x \| = \operatorname { dist } ( x , \mathbf{Z} ) = | x - N ( x ) | denotes the distance from x to the nearest integer.) It follows that H is countable. The set E also contains the set S of Pisot numbers (cf. Pisot number) and the set T of Salem numbers (cf. Salem number).
The recurrent E-sequences are those that satisfy linear recurrence relations. The corresponding subset of E is denoted by E_r. It was shown in [a6] that E _ { r } = S \cup T. A proof that E = E_r, as envisaged in [a7], would show that E = S \cup T and hence that:
i) H = S; and
ii) T is dense in [ 1 , \infty ). However, it was proved in [a2] that there are non-recurrent E-sequences and that the set of \theta ( a _ { 0 } , a _ { 1 } ) corresponding to these is dense in [ ( 1 + \sqrt { 5 } ) / 2 , \infty ). While this does not settle the question of whether E = E_r (since a given \theta might arise from both a recurrent and a non-recurrent Pisot sequence) it makes this unlikely. The prevailing opinion is that i) is true (Pisot's conjecture), but that ii) is false.
Families of E-sequences of the type E ( a _ { 0 } , c _ { 1 } + a _ { 0 } ^ { 2 } m ) were studied in [a5], where conditions are given under which each member of such a family will satisfy a linear recurrence for sufficiently large m. In this case the degree of the recurrence does not depend on m. For example, E ( 7,49 m + 15 ) is recurrent for m \geq 1 [a5] but is non-recurrent for m = 0 [a3].
Many generalizations of Pisot sequences are possible and some were already considered by Ch. Pisot in [a3] (see also [a1], Chapts. 13; 14). One interesting variant replaces the rounding operator N ( x ) by other operators, perhaps dependent on n. This can have a dramatic affect on the possible linear recurrence relations satisfied by the sequences (see, e.g. [a4]).
References
[a1] | M.J. Bertin, A. Decomps–Guilloux, M. Grandet–Hugot, M. Pathiaux–Delefosse, J.P. Schreiber, "Pisot and Salem Numbers" , Birkhäuser (1992) |
[a2] | D.W. Boyd, "Pisot sequences which satisfy no linear recurrence" Acta Arith. , 32 (1977) pp. 89–98 (See also: vol. 48 (1987), 191-195) |
[a3] | D.W. Boyd, "Pisot and Salem numbers in intervals of the real line" Math. Comp. , 32 (1978) pp. 1244–1260 |
[a4] | D.W. Boyd, "Linear recurrence relations for some generalized Pisot sequences" F.Q. Gouvea (ed.) N. Yui (ed.) , Advances in Number Theory , Oxford Univ. Press (1993) pp. 333–340 |
[a5] | D.G. Cantor, "On families of Pisot E-sequences" Ann. Sci. Ecole Norm. Sup. , 9 : 4 (1976) pp. 283–308 |
[a6] | P. Flor, "Über eine Klasse von Folgen naturlicher Zahler" Math. Ann. , 140 (1960) pp. 299–307 |
[a7] | Ch. Pisot, "La répartition modulo 1 et les nombres algébriques" Ann. Scuola Norm. Sup. Pisa Cl. Sci. , 7 : 2 (1938) pp. 205–248 |
Pisot sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pisot_sequence&oldid=55318