A Log-Log Saving for Matrix-Algebra Length and Terseness

Let $\mathrm{\ell }({\text{Mat}}_n(F))$ denote the length of the full matrix algebra for a field $F$, i.e. the largest of the least word length needed to span ${\text{Mat}}_n(F)$, over all generating sets $S$ of ${\text{Mat}}_n(F)$. Šitov proved the general estimate $$\mathrm{\ell }({\text{Mat}}_n(F))\le 2n{\log }_{2}n+4n-4.$$ The purpose of this paper is to obtain a log-log saving, and prove that for every $n>1$, $$\mathrm{\ell }({\text{Mat}}_n(F))\le 2n{\log }_{2}n-2n{\log }_2{\log }_{2}n+5n\mathrm{.}$$ A theorem of Specht gives a word-criterion for unitary similarity of complex $n\times n$ matrices. The trace argument of Freedman–Gupta–Guralnick, as used by Pappacena, shows that any upper bound on $\mathrm{\ell }({\text{Mat}}_n(F))$ can be used to bound the \emph{terseness} $\tau (n)$, i.e. the least upper bound for the length of words needed in Specht’s theorem. Thus, for $n>1$, $$\tau (n)\le 4n{\log }_{2}n-4n{\log }_2{\log }_{2}n+10n+1.$$