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 $\{\lim_{n\to\infty} f_n(\alpha) : \alpha\in\mathbf{Q}\}$?