Let $u_n = x + n\alpha \pmod 1$ with $\alpha$ irrational. We know that $(u_n)_{n \geq 0}$ is dense in $\mathbb{R}/\mathbb{Z}$ (equivalently $(u_n)_{n \geq 0}$ visits every open interval infinitely many times).

Now, let $(\epsilon_n)_{n \geq 0}$ be a sequence of independent uniform Bernoulli random variables on $\{0,1\}$, and define the random integers $K_n=\sum_{i = 0}^n \epsilon_i2^i$. Is it true that the random sequence $(u_{K_n})_{n \geq 0}$ almost surely visits every open interval infinitely many times ?

As a conditional and secondary question, if it is true, I would also like to know:

is it more generally true for random integers $K_n$ defined in the same way but using another representation of the integer numbers instead of the binary one ?

is it more generally true for the orbits of a minimal homeomorphism ? (after replacing intervals with open sets)