## Abstract

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

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.

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Communicated by J. Schoißengeier.

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Stankov, D. On spectra of neither Pisot nor Salem algebraic integers.
*Monatsh Math* **159**, 115–131 (2010). https://doi.org/10.1007/s00605-008-0048-0

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DOI: https://doi.org/10.1007/s00605-008-0048-0