On the intersection of Cantor set with the unit circle and some sequences

For $\lambda \in (0,1\mathrm{/}2)$ let $K_{\lambda }$ be the self-similar set in $\mathbb{R}$ generated by the iterated function system $\{f_0(x)=\lambda x,f_1(x)=\lambda x+1-\lambda \}$. In this paper, we investigate the intersection of the unit circle $\mathbb{S}\subset {\mathbb{R}}^2$ with the Cartesian product $K_{\lambda }\times K_{\lambda }$. We prove that for $\lambda \in (0,2-\sqrt{3}]$, the intersection is trivial, i.e., $$\mathbb{S}\cap (K_{\lambda }\times K_{\lambda })=\{(0,1),(1,0)\}\mathrm{.}$$ If $\lambda \in [0.330384,1\mathrm{/}2)$, then the intersection $\mathbb{S}\cap (K_{\lambda }\times K_{\lambda })$ is non-trivial. In particular, if $\lambda \in [0.407493,1\mathrm{/}2)$ the intersection $\mathbb{S}\cap (K_{\lambda }\times K_{\lambda })$ is of cardinality continuum. Furthermore, the bound $2-\sqrt{3}$ is sharp: there exists a sequence $\{{\lambda }_n{\}}_{n\in \mathbb{N}}$ with ${\lambda }_n\searrow 2-\sqrt{3}$ such that $\mathbb{S}\cap (K_{{\lambda }_n}\times K_{{\lambda }_n})$ is non-trivial for all $n\in \mathbb{N}$. 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. By employing tools from number theory, including the quadratic reciprocity law, we analyze the intersection of Cantor sets with some sequences. A dichotomy is established in terms of the Legendre symbol associated with the digit set, revealing a fundamental arithmetic constraint governing such intersections.