Recall that Pisot numbers are algebraic integers greater than $1$, whose other Galois conjugates have modulus $<1$. The set of Pisot numbers is usually denoted $S$. It is known that $S$ is denumerable and closed (Salem). Alors, the sequence of derived sets $S',S'', ...$ does not terminate. The smallest accumulation point of $S$ is the golden ratio $\phi$.
Let $\lambda$ be an accumulation point in $S$. Therefore it is the limit of a sequence of Pisot numbers $\mu_n\ne\lambda$. It is clear that the degree of $\mu_n$ tends to $+\infty$, whereas its norm remains bounded as $n\rightarrow+\infty$. My question is about the Galois conjugates of $\mu_n$; most of them must be close to the unit circle, because their product equals $N(\mu_n)/\mu_n$. Given a Galois conjugate $\tau$ of $\lambda$, does there exist a Galois conjugate $\tau_n$ of $\mu_n$ such that $\tau_n\rightarrow\tau$ ? What is the statistics of the Galois conjugates of $\mu_n$ as $n\rightarrow+\infty$ ?
Example: the multinacci number of degree $d$ is the root $a_d>1$ of the polynomial $X^d-X^{d-1}-\cdots-X-1$. When $d\rightarrow+\infty$, one has $a_d\rightarrow2$. Then I have a proof that the empirical measure $$m_d:=\frac1d\sum_{a\sim a_d}\delta_a$$ where the sum runs over all the Galois conjugates of $a_d$ (including itself) converges vaguely towards the uniform measure over the unit circle.
Edit. The same property holds true in the situation depicted by Dufresnoy and Pisot. Let $P\in{\mathbb Z}[X]$ be the minimal polynomial of $\lambda$, unitary. Let $A\in{\mathbb Z}[X]$ be such that $|A(z)|\le|P(z)|$ on $\mathbb T$, the equality arising only at finitely many points. Define the polynomials $P_n^\pm(X)=X^nP(X)\mp A(X)$. For infinitely many pairs $(n,\pm)$, $P_n^\pm$ has only one root $\lambda_n^\pm$ away from the unit disk, and none over $\mathbb T$. This Pisot number tends to $\lambda$, and the empirical measure defined as above tends to $\frac1{2\pi}d\theta$. It is thus tempting to conjecture that in every situation, the empirical measure tends to the uniform measure.
