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General Forms for Minimal Spectral Values For a Class of Quadratic Pisot Numbers
Author(s) -
Borwein Peter,
Hare Kevin G.
Publication year - 2003
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609302001455
Subject(s) - mathematics , combinatorics , zero (linguistics) , integer (computer science) , polynomial , spectrum (functional analysis) , discrete mathematics , quadratic equation , mathematical analysis , physics , geometry , quantum mechanics , computer science , programming language , philosophy , linguistics
This paper studies the spectrum that results when all height one polynomials are evaluated at a Pisot number. This continues the research theme initiated by Erdős, Joó and Komornik in 1990. Of particular interest is the minimal non‐zero value of this spectrum. Formally, this value is denoted as l 1 ( q ), and this definition is extended to all height m polynomials asl m ( q ) ≔ inf ( | y | : y = ɛ 0 + ɛ 1 q 1 + … + ɛ n q n , ɛ i ∈ Z , | ɛ i | ⩽ m , y ≠ 0 ) .A recent result in 2000, of Komornik, Loreti and Pedicini gives a complete description of l m ( q ) when q is the Golden ratio. This paper extends this result to include all unit quadratic Pisot numbers. A main theorem is as follows. T HEOREM . Let q be a quadratic Pisot number that satisfies a polynomial of the form p ( x ) = x 2 − ax ± 1, with conjugate r. Let q have convergents { C k / D k } and let k be the maximal integer such that| D k r − C k | ⩽ m 1 1 − | r |;thenl m ( q ) = | D k q − C k | .A value related to l ( q ) is a ( q ), the minimal non‐zero value when all ±1 polynomials are evaluated at q . Formally, this is a ( q ) ≔ inf ( | y | : y = ɛ 0 + ɛ 1 q 2 + … + ɛ n q n , ɛ i = ± 1 , y ≠ 0 ) .An open question concerning how often a ( q ) = l ( q ) is also answered in this paper.