Abstract
In this chapter, we examine spectra, the sets of values which result when various classes of polynomials are evaluated at a fixed number q. When this class is <Emphasis FontCategory=“NonProportional”>F</Emphasis> and q is a Pisot number, the spectrum
is, quite surprisingly, discrete. Indeed, from El of Chapter 3, we have that for q a Pisot number and p ∈ <Emphasis FontCategory=“NonProportional”>Z</Emphasis> of height h with q not a root of p,
where the positive constant c(q, h) depends only on q and h. This suggests the question of establishing the exact value for c(q,h). Specifically, we search for the minimum positive value in the spectrum of height h polynomials evaluated at a number q, where q is between 1 and 2.
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Selected References
P. Borwein and K.G. Hare, Some computations on the spectra of Pisot and Salem numbers, Math. Comp, (to appear).
P. Borwein and K.G. Hare, General forms for minimal spectral values for a class of quadratic Pisot numbers, J. London Math. Soc. (to appear).
P. Erdős, I. Joó, and V. Komornik, Characterization of the unique expansions <Inline>1</Inline> and related problems, Bull. Soc. Math. France 118 (1990), 377–390.
I. Joó and F.J. Schnitzer, On some problems concerning expansions by noninteger bases, Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 133 (1996), 3–10.
Y. Peres and B. Solomyak, Approximation by polynomials with coefficients ±1, J. Number Theory 84 (2000), 185–198.
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© 2002 Springer-Verlag New York, Inc.
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Borwein, P. (2002). Spectra. In: Computational Excursions in Analysis and Number Theory. CMS Books in Mathematics / Ouvrages de mathématiques de la SMC. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21652-2_16
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DOI: https://doi.org/10.1007/978-0-387-21652-2_16
Publisher Name: Springer, New York, NY
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