The following is taken from [a post by Terence Tao on the Chowla conjecture and the Sarnak conjecture](https://terrytao.wordpress.com/2012/10/14/the-chowla-conjecture-and-the-sarnak-conjecture/):


Given a bounded sequence ${f: {\bf N} \rightarrow {\bf C}}$, define the topological entropy of the sequence to be the least exponent ${\sigma}$ with the property that for any fixed ${\varepsilon > 0}$, and for ${m}$ going to infinity the set ${\{ (f(n+1),\ldots,f(n+m)): n \in {\bf N} \} \subset {\bf C}^m}$ of ${f}$ can be covered by ${O( \exp( \sigma m + o(m) ) )}$ balls of radius ${\varepsilon}$ (in the ${\ell^\infty}$ metric). A sequence is called deterministic of the topological entropy defined as above is zero.


Based on Tao's post, the sequence $\{e^{2\pi i \alpha n^2}\}\_{n\in \bf N}$ is deterministic. His argument is:



> 
> the ${m}$-blocks of such polynomials sequence have covering numbers that only grow polynomially in ${m}$, rather than exponentially, thus yielding the zero entropy.
> 
> 
> 


But this argument is a little heuristic to me: It is hard for me to find a bound for the covering number. In fact, what we need to do is that for any $\sigma >0, \varepsilon >0 $ the sequence


$$S(m,f)=\{(e^{2 \pi i \alpha (n+1)^2}, \dots, e^{2 \pi i \alpha (n+m)^2)}): n\in \bf N\}$$


can be covered by ${O( \exp( \sigma m + o(m) ) )}$ balls of radius ${\varepsilon}$. I personally believe that the omega notations could involve $\varepsilon$, please let me know if I am wrong.


Based on Tao's argument, we can actually make $S(m,f)$ covered by $P(m,\varepsilon)$-many balls of radius $\varepsilon$ (in ${\ell^\infty}$ metric) where $P(m,\varepsilon)$ is a polynomial in $m$ whose coefficients might depend on $\varepsilon$. But how to find such a cover exactly?



An example of deterministic sequence from Terence Tao's blog
Let $S\_1$ be a subset of $[0,1]$ with consecutive elements separated by a distance of at most $\epsilon/2\pi $ and let $S\_2$ be a subset of $[0,1]$ with consecutive elements separated by at most $\epsilon / (2 \pi m)$.


We can take $|S\_1| < 2\pi\epsilon^{-1} +1$ and $|S\_2| < 2\pi m \epsilon^{-1} +1$.


Let $\gamma \in S\_1$ be the closest point to $\alpha n^2 \mod 1$ and let $\beta \in S\_2$ be the closest point to $2 \alpha n \mod 1$. Then for all $j$ from $1$ to $m$


$$ \left| e^{ 2\pi i \alpha(n+j)^2} -e^{ 2\pi i (\alpha j^2 + \beta j + \gamma) } \right| <2 \pi \left|\alpha(n+j)^2 -(\alpha j^2 + \beta j + \gamma)\right| = 2\pi \left| \alpha n^2- \gamma + 2 \alpha n j - \beta j + \alpha j^2 - \alpha aj^2 \right| \leq |\alpha n^2 - \gamma| + j |2\alpha n - \beta| \leq \frac{\epsilon}{2} + \frac{j \epsilon}{2m } \leq \epsilon $$


giving a $\delta$-covering of size


$$|S\_1| |S\_2| \leq( 2\pi\epsilon^{-1} +1 )( 2\pi m \epsilon^{-1} ). $$


This is indeed a polynomial in $m$ with coefficients depending on $\epsilon$ (and in fact is polynomial in $\epsilon$).


