We investigate the class of $\pm1$ polynomials evaluated at $q$ defined as: \[ A(q)=\{\epsilon_0+\epsilon_1q+\cdots+\epsilon_m q^m :\epsilon_i\in\{-1,1\}\} \] and usually called spectrum, and show that, if $q$ is the root of the polynomial $x^n-x^{n-1}-\dots-x^{k+1}+x^k+x^{k-1}+\cdots+x+1$ between 1 and 2, and $n>2k+3$, then $A(q)$ is discrete, which means that it does not have any accumulation points.