Fermat's Last Theorem in star-invariant subspaces: capacity-zero boundary spectrum

Abstract

We prove that the Fermat equation $f^n+g^n=h^n$ with exponent $n\ge 4$ has no nontrivial solutions $f,g,h$ in the star-invariant subspace $K_{\theta }^p$, in which $1\le p\le \infty$, whenever the boundary spectrum of the inner function $\theta$ has logarithmic capacity zero. This gives a positive answer, for this class of inner functions and exponents, to a problem posed by Dyakonov.

6 pages