On the frequency of height values

We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height $${\mathcal {H}}$$ H with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to $${\mathcal {H}}$$ H 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 $$k \in \{0,d\}$$ k ∈ { 0 , d } or $$\gcd (k,d) = 1$$ gcd ( k , d ) = 1 . We therefore study the behaviour in the case where $$0< k < d$$ 0 < k < d and $$\gcd (k,d) > 1$$ gcd ( k , d ) > 1 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.