On the frequency of height values

We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height [Formula: see text] with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to [Formula: see text] that the height assumes on algebraic numbers of degree d with precisely k conjugates that lie inside the open unit disk. For both counts, we do not obtain an asymptotic, but only a rough order of growth, which arises from an asymptotic for the logarithm of the counting function; for the first count, even this rough order of growth exists only if [Formula: see text] or [Formula: see text] . We therefore study the behaviour in the case where [Formula: see text] and [Formula: see text] in more detail. We also count integer polynomials of fixed degree and fixed Mahler measure with a fixed number of complex zeroes inside the open unit disk (counted with multiplicities) and study the dynamical behaviour of the height function.