  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.
