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 $n\geq 4$ has no nontrivial solutions $f,g,h$ in the star-invariant subspace $K_θ^p$, in which $1\leq p\leq\infty$, whenever the boundary spectrum of the inner function $θ$ has logarithmic capacity zero. This gives a positive answer, for this class of inner functions and exponents, to a problem posed by Dyakonov.

Disclosure

“acknowledge fruitful discussions with ChatGPT, which located crucial references which finally permitted the author to make some progress on this problem after many years of unsuccessful contemplation. After the initial draft was complete, Claude was then used to proofread and criticize this note. Any remaining mistakes are, of course, the fault of the author. This brief note is organized as follows. Section 2 contains preliminary material about the projective Fermat curve and h”

PDF page 3

Classification

Literature search

Multiplier

2

Verified

2026-08-23T05:11:18.591487Z

Structural counts