Pisot sequence

The standard Pisot $E$-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