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

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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

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