Fractal representations of the number zero on the parabola curve

Motivated by several results in the study of unique q-expansions, this paper investigates the following problem. Let $K\subset R^2$ be a self-similar set with the convex hull ${[0,1]}^2$. How many distinct pairs $(x,y)\in K$ satisfy the equation$0=y-x^2?$ We establish the following result:

For any $\alpha \in (0,2)$ and any $ϵ>0$, there exists a homogeneous self-similar set K (with convex hull ${[0,1]}^2$) such that$\alpha -ϵ<{\dim }_H\left(K\right)<\alpha ,$ and the equation$0=y-x^2,(x,y)\in K,$ has exactly countably many distinct solutions. Specifically,$\left\{\right(x,y):y=x^2\}\cap K=\{(\frac{1}{m^k},\frac{1}{m^{2k}}):k\in N^{+}\cup \{0\left\}\right\}\cup \left\{\right(0,0\left)\right\},$ where ${\dim }_H$ denotes the Hausdorff dimension, and $1/m$, $m\in N^{+}$, represents the similarity ratio of K. Similar result can be proved for the Bedford-McMullen carpet.