摘要:
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 Erdos, Joo 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 epsilon(0) + epsilon(1)q(1) +... + epsilon(n)q(n), epsilon(i) is an element of Z, \epsilon(i)\ less than or equal to m, y not equal 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.THEOREM. 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\ less than or equal to 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 isa(q) := inf(\y\ : y = epsilon(0) + epsilon(1)q2 + ...+ epsilon(n)q(n), epsilon(i) = +/-1, y not equal 0).An open question concerning how often a(q) = l(q) is also answered in this paper.