论文信息 - Some computations on the spectra of Pisot and Salem numbers

Some computations on the spectra of Pisot and Salem numbers

Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erdos. Joo and Komornik in 1990. is the determination of l(q) for Pisot numbers q, where l(q) = inf(|y|:y = e0 + e1q1 + ...+ enqn, ei ∈ {±1,0}, y ≠ 0.) Although the quantity l(q) is known for some Pisot numbers q, there has been no general method for computing l(q). This paper gives such an algorithm. With this algorithm, some properties of l(q) and its generalizations are investigated.A related question concerns the analogy of l(q), denoted a(q), where the coefficients are restricted to ±1; in particular, for which non-Pisot numbers is a(q) nonzero? This paper finds an infinite class of Salem numbers where a(q) ≠ 0.

Peter B. Borwein | Kevin G. Hare | P. Borwein | K. Hare

[1] A. Garsia. Arithmetic properties of Bernoulli convolutions , 1962 .

[2] Vilmos Komornik,et al. On the sequence of numbers of the form $ε₀ + ε₁q + ... + ε_nq^n$, $ε_i ∈ {0,1}$ , 1998 .

[3] Henri Cohen,et al. A course in computational algebraic number theory , 1993, Graduate texts in mathematics.

[4] Toufik Zaïmi. On an approximation property of Pisot numbers , 2004 .

[5] K. Lau. Dimension of a Family of Singular Bernoulli Convolutions , 1993 .

[6] Y. Bugeaud. On a property of Pisot numbers and related questions , 1996 .

[7] V. Komornik,et al. An approximation property of Pisot numbers , 2000 .

[8] Vilmos Komornik,et al. Characterization of the unique expansions $1=\sum^{\infty}_{i=1}q^{-n_ i}$ and related problems , 1990 .

[9] Maurice Mignotte,et al. Mathematics for computer algebra , 1991 .

[10] R. Salem,et al. Power series with integral coefficients , 1945 .

[11] P. Erdös,et al. On a problem of Tamas Varga , 1992 .

[12] László Lovász,et al. Factoring polynomials with rational coefficients , 1982 .

[13] David W. Boyd. Pisot and Salem numbers in intervals of the real line , 1978 .

[14] M. Newman,et al. Topics in Algebra , 1978 .

[15] David W. Boyd,et al. On beta expansions for Pisot numbers , 1996, Math. Comput..

[16] Paul Erdös,et al. Developments in Non-Integer Bases , 1998 .