Equidistribution conjecture for the class and regulator data in the Greenberg algorithm

Fix $n\gg0$. Let ${\mathfrak t}_j$ be the fundamental ideals encountered by the algorithm that inductively computes the successive class and norm factors in $b_n$ steps; when ${\mathfrak t}_j$ is principal, write ${\mathfrak t}_j=(\tau_j)$. Let $c\!\ell_k({\mathfrak t}_j)$ denote the ideal class of ${\mathfrak t}_j$ in ${\mathcal C}_k$, let $\iota$ be the relevant embedding, let $\overline E_k$ be the closure of the unit group, and let $U_k^*$ be the group of local units. Define

$$\log({\mathfrak t}_j):=\log(\iota\tau_j)\pmod{\log(\overline E_k)},\qquad {\mathcal R}_k=\log(U_k^*)/\log(\overline E_k).$$

Equidistribution conjecture. The following assertions hold:

1. The classes $c\!\ell_k({\mathfrak t}_j)$ are uniformly distributed in ${\mathcal C}_k$.
2. Whenever ${\mathfrak t}_j=(\tau_j)$, the images $\log({\mathfrak t}_j)$ are uniformly distributed in the normalized regulator ${\mathcal R}_k$.

The conjecture is motivated by treating the ideals produced by the algorithm as random and is intended to explain why the algorithm should avoid unbounded behavior. The source does not provide a resolution or a precise probabilistic formulation beyond these uniform-distribution assertions.