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1. What the question is asking (in plain language)

For a group word

[ w=w(x_{1},\dots ,x_{n}) ]

let

[ \operatorname{Prob}{G}(w)= \frac{\bigl|{(a{1},\dots ,a_{n})\in G^{n}\;:\;w(a_{1},\dots ,a_{n})=1}\bigr|} {|G|^{\,n}} ]

be the probability that a randomly chosen (n)-tuple of elements of a finite group (G) satisfies the identity (w=1).

The question is:

Is it true that for every non‑trivial word (w) there is a constant (\varepsilon(w)>0) such that [ \operatorname{Prob}_{G}(w)>1-\varepsilon(w)\quad\Longrightarrow\quad w\ \text{holds identically in }G\ ? ]

The converse direction is trivial (if the identity holds in (G) then the probability is (1)).
Thus the problem asks whether every non‑trivial word has a positive gap: if the word is “

Original question: Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way? on Mathematics Stack Exchange, licensed CC BY-SA.