The Cubic Fermat Equation in Model Spaces: A Negative Answer to a Question of Dyakonov

For an inner function \(\theta\) on the unit disc and \(1\le p\le\infty\), let \(K^p_\theta=H^p\cap\theta\,\overline{H^p_0}\) be the associated model (star-invariant) subspace of the Hardy space \(H^p\). K. M. Dyakonov asked (Oberwolfach Reports 7 (2010); Integral Equations Operator Theory 78 (2014)) whether “Fermat's last theorem” holds in these spaces: does \(f^n+g^n=h^n\) with \(f,g,h\in K^p_\theta\) and \(n\ge3\) force \(f\), \(g\), \(h\) to lie in a one-dimensional subspace? Dyakonov noted that the answer is yes when \(\theta\) is a finite Blaschke product, and S. R. Garcia (2026) proved that it is yes for \(n\ge4\) when the boundary spectrum of \(\theta\) has logarithmic capacity zero, and for \(n=3\) when the boundary spectrum has at most one point. We show that the answer is no for \(n=3\). We construct an explicit inner function \(\theta_0=zB\widetilde B\), where \(B\) and \(\widetilde B\) are Blaschke products whose zeros accumulate on the unit circle exactly at \(-1\) and \(1\), and functions \(F,G,H\in K^\infty_{\theta_0}\), hence in \(K^p_{\theta_0}\) for every \(p\), such that \(F^3+G^3=H^3\) and \(F\), \(G\), \(H\) are linearly independent; moreover \(F+iH=B\), and the inner factors of \(F\), \(G\), \(H\) are pairwise coprime. The construction composes Dixon's elliptic functions \(\operatorname{sm}\) and \(\operatorname{cm}\) with a conformal map of the disc onto a period strip on whose two boundary lines both functions are real. Consequently the one-point hypothesis in Garcia's cubic theorem cannot be relaxed to two points, and every closed subset of the unit circle with at least two points is the boundary spectrum of an inner function \(\theta\) for which \(K^p_\theta\) contains a nontrivial solution of the cubic Fermat equation. The case \(n\ge4\) and Dyakonov's companion question on extreme points are not addressed. This is an unrefereed note.