On a limiting fractal measure defined by conjugate algebraic integers. (English) Zbl 0692.12002

Davie, A. M.; Smyth, C. J.

Publ. Math. Orsay 89/01, 93-103 (1989).

The authors define a sequence \((\beta_ i)_{i=0,1,2,...}\) of real algebraic integers via \(\beta_ 0=1\) and \(\beta_ i\) being the positive root of \(\beta_ i-\beta_ i^{-1}-\beta_{i-1}=0\). They discuss the distribution of the conjugates \(\beta_ i^{(j)}\) of \(\beta_ i\) on the real line and prove that the sequence \[ (\prod^{\deg (\beta_ i)}_{j=1}\max (1,\quad | \beta_ i^{(j)}|^ 2))^{1/\deg (\beta_ i)} \] is monotonously increasing.
Thus they extend and improve on earlier results of the second author [J. Aust. Math. Soc., Ser. A 30, 137-149 (1980; Zbl 0457.12001), and Math. Comput. 37, 205-208 (1981; Zbl 0475.12001)].

Reviewer: M.Pohst

MSC:

11R04	Algebraic numbers; rings of algebraic integers
12-04	Software, source code, etc. for problems pertaining to field theory

Keywords:

fractal measure; measures of algebraic integers; distribution of conjugates of special algebraic integers; real algebraic integers

Citations:

Zbl 0457.12001; Zbl 0475.12001