Exact overlaps on the projections of the four corner Cantor set

The four corner Cantor set $C$ is a planar self-similar set generated by the IFS $\{(\frac{x}{4}, \frac{y}{4}),(\frac{x+3}{4}, \frac{y}{4}),(\frac{x}{4}, \frac{y+3}{4}), (\frac{x+3}{4}, \frac{y+3}{4})\}$. In this paper we show that for $t>0$ its projection $C_t:=\{x+ty: (x,y)\in C\}$ is a self-similar set having an exact overlap if, and only if, $t=p/q\in\mathbb Q$ in lowest term satisfies $p\notin\Gamma$ and $q\notin\Gamma$, where $\Gamma:=\{(2k-1)2^{2\ell-1}: k,\ell\in\mathbb N\}$. Let $W$ be the set of all coprime pairs $(p,q)\in\mathbb N^2$ such that $C_{p/q}$ has an exact overlap. By using properties of Euler function from analytic number theory we show that \[ \lim_{N\to\infty}\frac{\#(W\cap[1,N]^2)}{N^2}=\frac{10}{3\pi^2}. \]Comment: 21 pages, 4 figure