On the intersection of Cantor sets with the unit circle and some sequences
Abstract
Abstract For , let be the self‐similar set in generated by the iterated function system . In this paper, we investigate the intersection of the unit circle with the Cartesian product . We prove that for , the intersection is trivial , that is, If , then the intersection is nontrivial. In particular, if , the intersection is of cardinality continuum. Furthermore, the bound is sharp: there exists a sequence with such that is nontrivial for all . This result provides a negative answer to a problem posed by Yu (2023). Our methods extend beyond the unit circle and remain effective for many nonlinear curves. We also characterize the intersection of missing digits Cantor sets with the sequence by utilizing the Legendre symbol.
Fields of Study & Subjects
Mathematical Dynamics and Fractals
Holomorphic and Operator Theory
Nonlinear Partial Differential Equations
Publication details
- Journal / source
- Journal of the London Mathematical Society
- Publisher
- Wiley
- Volume
- 113
- Issue
- 1
- Publication date
- 1 January 2026
- Language
- EN