On totally real numbers and equidistribution

C.J. Smyth and later Flammang studied the spectrum of the Weil height in the field of all totally real numbers, establishing both lower and upper bounds for the limit infimum of the height of all totally real integers and determining isolated values of the height. We remove the hypothesis that we consider only integers and establish an lower bound on the limit infimum of the height for all totally real numbers. Our proof relies on a quantitative equidistribution theorem for numbers of small height.