On spectra of neither Pisot nor Salem algebraic integers

Abstract

We investigate the class of  ± 1 polynomials evaluated at a real number q> 1 defined as:

$$A(q)=\{\epsilon_0+\epsilon_1q+\cdots+\epsilon_k q^k : \epsilon_i\in\{-1,1\}\}$$

and usually called spectrum. Λ(q) is defined by analogy where the coefficients are extended to ± 1,0. In this paper an algorithm for finding the spectrum of a real algebraic integer q without a conjugate of modulus 1 is given. The algorithm can check and terminate if A(q) or Λ(q) is not discrete. Using this criterion a part of a conjecture of Borwein and Hare is proved: if 1 < q < 2 and A(q) is discrete, then all real conjugates of q are of modulus less than q. For an infinite sequence r _2m Zaimi has proved that A(r _2m ) is discrete, and here it is proved that Λ(r _2m ) is not discrete. For given ${q\in (1,2)}$ , an algorithm counting a sequence of  ± 1 polynomials P _n (x) such that, solutions α _n  > q of equations ${P_n(x)=\frac{1}{x-1}}$ satisfy α _n → q, is presented.