Discrete spectra and Pisot numbers

By the m-spectrum of a real number q>1 we mean the set Ym(q) of values p(q) where p runs over the height m polynomials with integer coefficients. These sets have been extensively investigated during the last fifty years because of their intimate connections with infinite Bernoulli convolutions, spectral properties of substitutive point sets and expansions in noninteger bases. We prove that Ym(q) has an accumulation point if and only if q