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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:

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

  2. 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.

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    $\begingroup$ The answer for 2 is yes and the proof is easy and follows easily from unique ergodicity of the Kronecker system. $\endgroup$ – Asaf Feb 23 '17 at 15:52
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As it is stated, question 1 has a negative answer: take any equidistributed sequence, and insert a sequence of $n$ $0$'s between $x_{2^n}$ and $x_{2^n+1}$. This will change the natural density of no set $A=A({\bf x},a,b)$, although it will make vanish the limit $\lim_{d\to\infty }\min_n{|A\cap\{n+1,\dots,n+d\}|\over d }.$

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Your questions is closely related to the concept of so-called well-distribution of sequences. A sequence $(x_n)_{n \geq 1} \in [0,1]$ is called well-distributed if $$ \lim_{n \to \infty} \frac{1}{N} \Big\{ 1 \leq k \leq n:~x_{m+k} \in [a,b] \Big\} = b-a $$ holds uniformly in $m$. Asaf's comment says that the Kronecker sequences is not only equidistributed, but even well-distributed. Many other results can also be carried over from uniform distribution theory. See for example: B. Lawton, A note on well distributed sequences. Proc. Amer. Math. Soc. 10, 1959, 891–893. He proves a variant of van der Corput's difference theorem, and of Weyl's theorem on the equidistribution of polynomial sequences.

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