# Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way?

Suppose for an arbitrary group word $$w$$ ower the alphabet of $$n$$ symbols $$\mathfrak{U_w}$$ is a variety of all groups $$G$$, that satisfy an identity $$\forall a_1, … , a_n \in G$$ $$w(a_1, … , a_n) = e$$. Is it true, that for any group word $$w$$ there exists a positive real number $$\epsilon (w) > 0$$, such that any finite group $$G$$ is in $$\mathfrak{U_w}$$ iff $$\frac{\lvert\{(a_1, … , a_n) \in G^n : w(a_1, … , a_n) = e\}\rvert}{{|G|}^n} > 1 - \epsilon(w)?$$

How did this question arise? There is a widely known theorem proved by P. Erdős and P. Turán that states:

A finite group $$G$$ is abelian iff $$\frac{|\{(a, b) \in G^2 : [a, b] = e\}|}{{|G|}^2} > \frac{5}{8}.$$

This theorem can be rephrased using aforementioned terminology as $$\epsilon([a, b]) = \frac{3}{8}$$.

There also is a generalisation of this theorem, stating that a finite group $$G$$ is nilpotent of class $$n$$ iff $$\frac{|\{(a_0, a_1, … , a_n) \in G^{n + 1} : [ … [[a_0, a_1], a_2]… a_n] = e\}|}{{|G|}^{n + 1}} > 1 - \frac{3}{2^{n + 2}},$$ thus making $$\epsilon([ … [[a_0, a_1], a_2]… a_n]) = \frac{3}{2^{n + 2}}$$.

However, I have never seen similar statements about other one-word varieties being proved or disproved, despite such question seeming quite natural . . .

Actually, I doubt that the conjecture in the main part of question is true. However, I failed to find any counterexamples myself.

• I one wrote out an answer which is about generalising the Erdos-Turan result to infinite groups: math.stackexchange.com/a/2809964/10513 You might find it interesting/relevant. Commented Jan 18, 2019 at 11:54
• Not mentioned in a 2015 survey Farrokhi, D. G. (2015). ON THE PROBABILITY THAT A GROUP SATISFIES A LAW: A SURVEY (Research on finite groups and their representations, vertex operator algebras, and algebraic combinatorics), muroran-it.ac.jp/mathsci/danwakai/past/articles/201404-201503/…. Mentioned as open in a note by John D. Dixon, "Probabilistic Group Theory", people.math.carleton.ca/~jdixon/Prgrpth.pdf
– Dap
Commented Jan 18, 2019 at 18:00
• Is the $n=1$ case obviously true? Or is even that case difficult? Commented Jan 23, 2019 at 15:48
• @MeesdeVries, for the case $n = 1$ only three results are currently known: $\epsilon(x) = \frac{1}{2}$, $\epsilon(x^2) = \frac{1}{4}$ and $\epsilon(x^3) = \frac{2}{9}$. Commented Aug 2, 2019 at 11:19
• I proved analogous results about the equations $xy^2=y^2x$, $xy^3=y^3x$ and $xy=yx^{-1}$ (and infinitely many others derived from these by adding further variables) earlier this year: preprint. Commented Oct 16, 2020 at 13:23