Solutions of $x^α+y^α=1$ in $λ$-Cantor sets

Abstract

In this paper, we study the structure of the solutions of $x^α+y^α=1$ with both coordinates in a $λ$-Cantor set. We prove that, for any $α>1$, the intersection with the middle-third Cantor set has positive Hausdorff dimension, strengthening the infinitude question posed by Jiang et al. [J. London Math. Soc. 113 (2026), e70408]. To explore the structure of the solutions, for arbitrary $α$, we find that their Hausdorff dimension undergoes phase transitions as $λ$ varies: we establish the occurrence of Hausdorff phase transitions by proving the existence of zero-dimensional and positive-dimensional regions.