Let $A_n$ be the set of all eigenvalues of binary matrices with trace $0$ (equivalently, the union of spectra of all simple digraphs on $n$ vertices), and let $A=\bigcup A_n$.

It is quite easy to prove that $A$ is dense in $\mathbb{C}$, but do we know anything about how it distributes?

In particular, given an $r$ do we know to bound $\lim_{n\to\infty} \frac{|A_n\cap B_r|}{|A_n|}$ as a function of $r$ (where $B_r=\{z\mid |z|\le r\}$)?


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