TITLE: Possible limits of $(1/n) \sum_{k=0}^{n-1} e^{i2\pi \cdot 2^k\alpha}$
QUESTION [22 upvotes]: I made a throwaway comment on math stackexchange the other day that got me thinking about the following question. Let
$$ f_n (\alpha) = \frac1n \sum_{k=0}^{n-1} e(2^k\alpha),$$
where $e(x) = \exp(i2\pi x)$.

Can you determine the set of possible limits $$ L = \{z : f_n(\alpha)\to z~\text{for some}~\alpha\}?$$

Of course by the ergodic theorem $f_n(\alpha)\to 0$ for almost every $\alpha$. Also $f_n(\alpha)\to 1$ for every dyadic rational, so for a comeagre set of $\alpha$ the sequence $f_n(\alpha)$ has a subsequence converging to $1$. So we're interested in a particular meagre null set of $\alpha$. :)
Here is everything I know about $L$:

$L$ is disjoint from a neighbourhood of $S^1\setminus\{1\}$. Indeed suppose $f_n(\alpha)\approx z$, where $|z|=1$. Then $e(2^k\alpha) \approx z$ for almost all $k\leq n$, so also $e(2^{k+1}\alpha)\approx z$ for almost all $k\leq n$, so $z^2 \approx z$, which implies $z\approx 1$.
$L$ is a closed, convex subset of $\{z: |z|\leq 1\}$. Hand-waving argument for convexity: Given $f_n(\alpha)\to z$ and $f_n(\beta)\to w$, with $\alpha,\beta\in[0,1]$, find $N$ such that $f_N(\alpha)\approx z$ and $f_N(\beta) \approx w$, write $\alpha_N$ and $\beta_N$ for $\alpha$ and $\beta$ rounded to the nearest multiple of $2^{-N}$, and then consider $$\gamma_N = \alpha_N + \beta_N/2^N + \alpha_N/2^{2N} + \beta_N/2^{3N} + \cdots.$$ Then the sequence $f_n(\gamma_N)$ hovers around the point $(z+w)/2$ for all sufficiently large $n$. Now patch the sequence $(\gamma_N)$ together in a similar way. Other convex combinations $pz + (1-p)w$ can be obtained similarly. Closedness is also similar.
Obviously $f_n(\alpha)$ converges for every rational $\alpha$, and the limits tend to be interesting. For example, the points $1,-1/2,(1\pm i\sqrt{7})/6$ are in $L$, and so also by 2 their convex hull is contained in $L$. Thus at least $L$ contains a neighbourhood of $0$. By continuing in this way we begin to see the following picture.


Most, but not all, of the points on the boundary appear to be obtained from $\alpha$ of the form $1/(2^n-1)$.

In fact $L = \{\int z\,d\mu : \mu~\text{a}~\times 2\text{-invariant measure on}~S^1\}.$ To see the inclusion $\subset$, take $\mu$ to be any weak* limit point of the sequence of measures $\frac1n \sum_{k=0}^{n-1} \delta_{2^k\alpha}$. To see the inclusion $\supset$, from the ergodic decomposition and the convex of $L$ observe that we may assume $\mu$ is ergodic for $\times 2$, and in this take $\alpha$ to be a $\mu$-random point and apply the ergodic theorem. This observation suggests the points $\int z \,d\mu_p$, where $\mu_p$ is the $\times 2$-invariant measure in which binary digits are independent and come up $1$ with probability $p$. However, this appears to constitute only a strict subset of the above picture, as shown below.


To me these observations strongly suggest the following question.

Is $L$ the convex hull of $\{\lim_{n\to\infty} f_n(\alpha) : \alpha\in\mathbf{Q}\}$?

REPLY [12 votes]: See the paper "Le poisson n'a pas d'arêtes" by Thierry Bousch. This is a joke that was explained to me much later. The set is considered to resemble a fish. The French word arête means both bone and edge (of a polygon); so the English title of the paper would be "The fish has no bones/The fish has no edges". The paper proves that every point on the boundary is an extreme point. (so that the answer to your question is No). The set was also studied in works of Oliver Jenkinson.