Elements Powering to Strong Involutions in Finite Classical Groups of Odd Characteristic: Order 1/log n and a Log-Periodic Asymptotic

Let \(G\) be a finite \(n\)-dimensional classical group over a field of odd order \(q\), and let \(Q\) be the set of elements \(g\) of even order whose involution \(g^{|g|/2}\) has a fixed space of dimension in \([n/3,2n/3)\). Lübeck, Niemeyer and Praeger proved \(|Q|/|G|\ge c/\log n\), and Praeger, with Lübeck and Niemeyer, asked at Oberwolfach in 2011 for the asymptotic value of \(|Q|/|G|\), in particular whether \(|Q|/|G|=O(1/\log n)\) (printed as \(O(\log n)\), evidently a misprint). For the groups \(X\) of the table of Lübeck, Niemeyer and Praeger and for \(\operatorname{Sp}_{2m}(q)\), \(\operatorname{SO}_{2m+1}(q)\) and \(\operatorname{SO}^\pm_{2m}(q)\) we prove \(|Q|/|X|\le C/\log n\) with absolute constants (\(C=100\) for \(\operatorname{GL}_n(q)\) and \(\operatorname{GU}_n(q)\); \(45\) for \(\operatorname{Sp}_{2m}(q)\), \(\operatorname{GSp}_{2m}(q)\), \(\operatorname{SO}_{2m+1}(q)\), \(\operatorname{GO}_{2m+1}(q)\); \(80\) for \(\operatorname{SO}^\pm_{2m}(q)\), \(\operatorname{GO}^\pm_{2m}(q)^0\)), uniformly in \(q\); with their lower bound, \(|Q|/|H|=\Theta(1/\log n)\) for every \(H\) between the groups \(S\le X\) of a line of the table, the upper constant carrying a factor at most \(q+1\) for the other \(H\), among them \(\operatorname{SL}_n(q)\) and \(\operatorname{SU}_n(q)\). For the eight groups listed we also prove \((\log n)|Q|/|X|=\Phi(\log_2\log n)+o(1)\) uniformly in \(q\), where \(\Phi(u)=\frac{\log 2}{3}\sum_{k\in\mathbb{Z}}2^{u-k}e^{-2^{u-k}/3}\) is \(1\)-periodic with mean \(1\), not constant, and within \(10^{-5}\) of \(1\). Thus \(|Q|/|X|\) is \(1/\log n\) up to a factor that tends to within \(10^{-5}\) of \(1\), while \((\log n)|Q|/|X|\) has no limit. The set \(Q\) depends only on semisimple parts, and the torus formula of Niemeyer and Praeger reduces the problem to random permutations, signed for the symplectic and orthogonal groups, whose cycles carry independent \(2\)-adic levels; what is added here is a level-by-level upper bound, the torus model for the groups of similitudes, and the asymptotic analysis of the levels. The full orthogonal groups \(\mathrm{O}^\pm_{2m}(q)\) are not covered; for \(\operatorname{SL}_n(q)\), \(\operatorname{SU}_n(q)\) and the groups strictly between \(S\) and \(X\) only the order of magnitude is proved, and for \(\operatorname{\Omega}\) only the upper bound; the convergence is slow, and the numerical values are evidence only. This is an unrefereed note.