Radial limit does not exist almost everywhere Problem 4 in Chapter 4 of Stein's book "Real Analysis" says 
$\sum_{n\geqslant 0}z^{2^n}$ 
doesn't have radial limit as $z$ approaches the unit circle from inside almost everywhere. It's fairly easy to find a dense set s.t. the radial limit doesn't exist ($=\infty$, actually), but is there a simple way to prove the set of divergence has positive measure?
 A: Put $f(z) = \sum_{n=0}^{\infty} z^{2^n}$.  First note that the set of $\theta \in [0,1)$ such that the binary expansion of $\theta$ has arbitrarily large strings of $100$ consecutive zeros is a set of measure $1$.  Now take such a $\theta$, and let $N$ be such that $\{2^{N} \theta\} \le 2^{-100}$ (here $\{x\}$ stands for the fractional part of $x$).  Consider $f(e^{-1/2^N + 2\pi i\theta})$ and $f(e^{-1/2^{N+50}+2\pi i \theta})$.  The difference of these two quantities is in size
$$ 
|f(e^{-1/2^N+2\pi i\theta}) - f(e^{-1/2^{N+50} +2\pi i\theta})|
$$ 
and using the triangle inequality appropriately this 
is 
$$ 
 \ge \Big|\sum_{n=N+1}^{N+50} e^{-2^n/2^{N+50}} e^{2\pi i 2^n \theta} \Big| - \sum_{n=0}^{N} (e^{-2^{n}/2^{N+50}} - e^{-2^n/2^N})  -  \sum_{n=N+51}^{\infty} e^{-2^{n}/2^{N+50}} -\sum_{n=N+1}^{\infty} e^{-2^n/2^N}. \tag{1} 
$$ 
The third term and fourth terms in (1) are together in size at most 
$$ 
2(e^{-2} + e^{-4} + \ldots )\le 1.  
$$
Since $|e^{-x}-e^{-y}| \le |x-y|$ for $x$ and $y$ in $[0,1]$, the second term in (1) is bounded in size by 
$$ 
\sum_{n=0}^{N} \Big( \frac{2^n}{2^N} - \frac{2^n}{2^{N+50}}\Big) \le 2.
$$
Finally, since $\{2^n \theta\} \le 2^{-50}$ for $N+1\le n \le N+50$, we see easily that the first term in (1) is 
$$
\ge \sum_{n=N+1}^{N+50} e^{-2^n/2^{N+50}} - \frac{1}{2^{20}} \ge 40. 
$$
It follows that the difference in the two values of $f$ is at least $30$ in size, so that the radial limit cannot exist for this $\theta$. 
A: I don't know any simply way, but I would be interested in one, too.
In fact $\sum_{n\geqslant 0}z^{2^n}$ has no radial limit anywhere on the unit circle. This follows from a 1928 Tauberian theorem of Ananda-Rau (see review here). The result is included as Theorem 104 in Hardy: Divergent series (Oxford Clarendon Press, 1948); the proof appears in the notes on Chapter VII.
For part (b) of the problem, see Theorem 6.4 in Chapter V (on Page 203) in Zygmund: Trigonometric Series I. Alternately, see Zygmund's original paper.
