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

TL;DR

This paper investigates the growth rates of totally real units and eigenvalue patterns in special linear and symplectic integer groups within thin tubes, revealing entropy formulas linked to root systems and providing bounds on conjugacy class counts.

Contribution

It introduces explicit entropy formulas for counting arithmetic objects in $ m{SL}_n(b Z)$ and $ m{Sp}_{2n}(b Z)$ along specific directions, connecting eigenvalue data to root system evaluations.

Findings

Growth rate of objects in thin tubes is exponential with rate given by entropy formulas.

Lower and upper bounds for conjugacy class counts are established, differing by a factor of two.

Entropy formulas are expressed via sums over positive roots of the associated Lie groups.

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 $\mathsf E_n(v)= \rho_{\operatorname{SL}_n}(v)=\sum_{i=1}^{n-1}(n-i)\,v_i,$ 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(\rho_{\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 $\exp\!\bigl(\rho_{\operatorname{SL}_n}(v)T\bigr)$ for the number of $\operatorname{SL}_n(\mathbb Z)$-conjugacy classes with a prescribed eigenvalue data; we also obtain an upper bound of order $\exp\!\bigl(2\rho_{\operatorname{SL}_n}(v)T\bigr)$. 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 $\mathsf E_{2n}^{\operatorname{Sp}}(v)=\rho_{\operatorname{Sp}_{2n}}(v)=\sum_{i=1}^n(n+1-i)v_i,$ the half-sum of positive roots of $\operatorname{Sp}_{2n}(\mathbb R)$.