A real sequence $\mathbf{x}=(x_n)_{n=1}^\infty\subseteq[0,1]$ is *equidistributed* if, for all $0\leq a< b\leq 1$,
$$
\lim_{n\to\infty}\frac{|\{1\leq k\leq n:x_k\in (a,b)\}|}{n}=b-a
$$

This can be restated as follows. Given $\mathbf{x}$, $a$, $b$ as above, let $$ A(\mathbf{x},a,b)=\{n\in\mathbb{N}:x_n\in (a,b)\}. $$ Then to say $\mathbf{x}$ is equidistributed is to say $A(\mathbf{x},a,b)$ has natural density $b-a$, for all $0\leq a< b\leq 1$.

I am interested in situations where something can be said about the *lower Banach density* of the sets $A(\mathbf{x},a,b)$. Specifically:

If $\mathbf{x}$ is equidistributed then is the lower Banach density of $A(\mathbf{x},a,b)$ positive?

If this is too much to ask, then I would restrict the question to the (equidistributed) sequences $\mathbf{x}=(\{n\alpha\})_{n=1}^\infty$, where $\{\cdot\}$ denotes fractional part and $\alpha$ is a fixed irrational number. Does the previous question have a positive answer for these sequences? Are there irrationals $\alpha$ such that the answer is positive?

EDIT: The accepted answer below (from Pietro Majer) gives a counterexample to Question 1. I thought it would be helpful for future users to detail (to the best of my understanding) Asaf's comment that Question 2 has a positive answer.

Fix an irrational $\alpha$ and let $T:S^1\longrightarrow S^1$ such that $T(z)=e^{2\pi i\alpha}z$. Then $T$ is Lebesgue invariant and uniquely ergodic by the Kronecker-Weyl Equidistribution Theorem. Fix $U\subseteq S^1$ open. For $n\geq 1$, let $f_n:S^1\longrightarrow[0,1]$ such that $$ f_n(z)=\frac{|\{1\leq k\leq n:T^k(z)\in U\}|}{n}. $$ By unique ergodicity, $(f_n)$ converges uniformly to (the constant function) $\lambda(U)$. Using this uniform convergence, it follows that the set $A(\mathbf{x},a,b)$ above (where $\mathbf{x}=(\{n\alpha\})_{n=1}^\infty$) has defined Banach density equal to $b-a$.

My source for the above is Katok & Hasselblatt, *Introduction to theTheory of Dynamical Systems*. The connection between unique ergodicity and Banach density is also explicitly mentioned in the footnotes on page 6 of this article by Downarowicz & Glasner. The slogan is: in a uniquely ergodic system every point is *uniformly generic*.