Counting totally real units and eigenvalue patterns in $\rm{SL}_n(\mathbb Z)$ and $\rm{Sp}_{2n}(\mathbb Z)$ in thin tubes

Abstract

For a vector $v=(v_1,\dots ,v_n)$ with $v_1>\cdots>v_n$ and $\sum v_i=0$, we study the "directional entropy" of two arithmetic objects: (1) the logarithmic embeddings of degree-$n$ totally real units, and (2) the logarithmic eigenvalue data of $\operatorname{SL}_n(\mathbb Z)$. In each case, the entropy in the direction of $v$ is $\mathsf E_n(v)= ρ_{\operatorname{SL}_n}(v)=\sum_{i=1}^{n-1}(n-i)\,v_i,$ the value of the half-sum of positive roots of $\operatorname{SL}_n(\mathbb R)$ evaluated at $v$. More precisely, the number of objects lying in a thin tube around the ray $\mathbb R_+v$ and of norm at most $T$ grows on the order of $ \exp\!\bigl(ρ_{\operatorname{SL}_n}(v)\,T\bigr)$ as $T\to \infty$. Because each eigenvalue data determines an $\operatorname{SL}_n(\mathbb R)$-conjugacy class, this implies a lower bound of order $\exp\!\bigl(ρ_{\operatorname{SL}_n}(v)T\bigr)$ for the number of $\operatorname{SL}_n(\mathbb Z)$-conjugacy classes with a prescribed eigenvalue data; we also obtain an upper bound of order $\exp\!\bigl(2ρ_{\operatorname{SL}_n}(v)T\bigr)$. A parallel argument for the symplectic lattice $\operatorname{Sp}_{2n}(\mathbb Z)$, taken in the symmetric direction $v=(v_1,\dots ,v_n,-v_n,\dots ,-v_1),\quad v_1>\cdots>v_n>0,$ shows that $\mathsf E_{2n}^{\operatorname{Sp}}(v)=ρ_{\operatorname{Sp}_{2n}}(v)=\sum_{i=1}^n(n+1-i)v_i,$ the half-sum of positive roots of $\operatorname{Sp}_{2n}(\mathbb R)$.