# Fraction of tuples for iterated Lie bracket word in subring of finite Lie ring is at least as much as in whole ring

## Statement

Suppose $w$ is a Lie ring word in $n$ letters that involves iterations of the Lie bracket only, and where every letter appears exactly once. For a finite Lie ring $R$, consider the fraction:

$f_w(R) := \frac{|\{(a_1,a_2,\dots,a_n) \in R^n \mid w(a_1,a_2,\dots,a_n) = 0\}|}{|R|^n}$

Then, if $S$ is a subring of $R$, we have:

$f_w(S) \ge f_w(R)$

The possibilities for $w$ include words such as $[[\dots[[x_1,x_2],x_3],\dots,x_{n-1}],x_n]$ (which define nilpotency class $\le n-1$ and words such as $[[x_1,x_2],[x_3,x_4]]$ (which defines derived length $2$).

## Facts used

1. Fraction of tuples satisfying groupy relation in subgroup is at least as much as in whole group