Spectral gap conjecture (Lubotzky–Phillips–Sarnak)

Let $\mu_2$ be the normalized Haar measure on $SU(2)$, and for $g = (g_1,\dots,g_n) \in SU(2)^n$ let $z_g$ be the Hecke operator $(z_g f)(x) = \sum_{j=1}^{n}\bigl(f(g_jx) + f(g_j^{-1}x)\bigr)$ on $L^2(SU(2),\mu_2)$.

For every integer $n > 2$, if $g_1,\dots,g_n$ are chosen independently and uniformly at random from $SU(2)$, then with probability $1$ the operator $z_g$ has a nontrivial spectral gap: for $\mu_2^{\otimes n}$-almost every $g \in SU(2)^n$,

This is the spectral gap conjecture. Combined with Fisher's zero–one law, it is equivalent to the existence of a spectral gap on a set of $n$-tuples of positive Haar measure.

Formalization Note. "Uniformly at random" is the product Haar measure on $SU(2)^n$; "spectral gap" is the quadratic-form condition from the definition file: some $\varepsilon>0$ with $\operatorname{Re}\langle z_g f,f\rangle \le (2n-\varepsilon)\|f\|_2^2$ for all mean-zero $f\in L^2$. The range $n > 2$ follows the source; the operator acts on $L^2(SU(2))$ as in the source, and a gap there implies one on $L^2(S^2)$.