Asymptotically optimal \(k\)-step nilpotency of quadratic algebras and the Fibonacci numbers (Q682120)

Applied to quadratic algebras, the Golod-Shafarevich theorem implies that an \(n\)-generated associative algebra with \(d\) quadratic relations cannot be \(k\)-step nilpotent if \(d<n^2\varphi_k\), where \(\varphi_k={1\over 4}\cos^{-2}\left({\pi\over k+1}\right)\). In the paper under review, the authors show that this estimate is asymptotically optimal. For every \(k\in\mathbb N\) they construct a sequence of \(n\)-generated algebras \(R_n\) with \(d_{nk}\) quadratic relations which are \(k\)-step nilpotent and \(\lim_{n\to\infty}{d\over n^2}=\varphi_k\). Curiously enough, for some pairs \((n,k)\) the estimate \(d_{nk}\) given in the paper is equal to the bound \(\varphi_k\) provided by the Golod-Shafarevich theorem. The authors illustrate this for \(k=4\) and \(k=5\) when \( \varphi_4={3-\sqrt{5}\over 2}\) and \( \varphi_5={1\over 3}\). Then show that \(d_{n4}=\lceil\varphi_4n^2\rceil\) if and only if \(n\) is a Fibbonacci number and \(d_{n5}=\lceil\varphi_5n^2\rceil\) if and only if \(n\in\{1,2\}\) or \(n\) is divisible by 6.