TITLE: Existence of radial limits of products of certain power series and $1-x$
QUESTION [6 upvotes]: Let $V$ be an arbitrary set of infinitely many positive integers, and let: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}z^{v}$$ Let $T_{V}$ denote the set of all $t\in\left[0,1\right)$ for which the limit:
$$c_{V}\left(t\right)\overset{\textrm{def}}{=}\lim_{x\uparrow1}\left(1-x\right)\varsigma_{V}\left(e^{2\pi it}x\right)$$
exists. 
Suppose $0\in{T_{V}}$ and $c_{V}\left(0\right)\neq0$ (that is, $V$ has a positive, well-defined natural density). Are there then any $t\in\left(0,1\right)$ which must be in $T_{V}$? More generally, what, if anything, can be said about $T_{V}$ and/or $\left[0,1\right)\backslash T_{V}$? (Ideally, all of $\mathbb{Q}$ (or all but some exceptional set) would have to be a subset of  $T_{V}$.) Conversely, what (if anything) does $\mathbb{Q}\subseteq T_{V}$ then force to be true about $V$? In particular, if $\varsigma_{V}\left(z\right)$ is a rational function, then $\mathbb{Q}\subseteq T_{V}$ necessarily holds. Thus, another way of phrasing the question is: does the set $\left\{ \varsigma_{V}\left(z\right):T_{V}\supseteq\mathbb{Q}\right\}$ contain functions which are not rational, and, if such functions do exist, what can be said about them and their associated $V$s?

REPLY [3 votes]: As for probabilistic approaches... let's just say they're not my cup of tea Well, that certainly has to be fixed if one "thinks of himself as an analyst". Let me give you a quick overview of the things most relevant to your current project as a series of exercises.
1) Read https://en.wikipedia.org/wiki/Hoeffding%27s_inequality 
2) Look up the proof of the Bernstein inequality
$$
\|P'\|_\infty\le Cn\|P\|_\infty
$$
for trigonometric polynomials $P(t)=\sum_{k=-n}^n c_ke^{ikt}$ of degree $n$. You do not need the sharp value of $C$.
3) Derive from it that $\|P\|_\infty\le C\max_{k=0}^{100n}|P(e^{(2\pi i k)/(100n)}|$ 
4) Using Hoeffding and 3), show that if $\xi_j$ are mean zero independent Bernoulli that are $p_j-1$ with probability $p_j$ and $p_j$ with probability $(1-p_j)$, then 
$$
P(\|\frac 1N\sum_{j=1}^N\xi_jz^j\|_\infty\ge N^{-0.1})\le N^{-0.1}
$$
for large $N$.
5) Using Borel-Cantelli and the fact that Chesaro means of a bounded sequence can be read from any subsequence $N_k$ with $N_{k+1}/N_k\to 1$ as $k\to\infty$, show that with probability $1$, we have
$$
\lim_{N\to\infty}\frac 1N \sum_{j=1}^N\xi_jz^j=0
$$ 
for all $z\in\mathbb T$ simultaneously.
6) Consider now the random set $V$ to which each particular integer $j$ belongs independently with probability $a_j\in[0,1]$. Show that with probability $1$, for every $t$,
$$
\lim_{N\to\infty}\frac 1N\left|\sum_{v\in V, v\le N}z^v-\sum_{v=1}^N a_vz^v\right|=0
$$ 
Thus, it does not matter if you consider sums over sets or arbitrary sums with coefficients in $[0,1]$, but with the latter you have way more freedom in various constructions. For instance,
7) Construct a series that gives you a non-zero limit at $0$, $z$, $\bar z$ but $0$ limit everywhere else.
8) Construct a series that gives you non-zero limits at any prescribed self-conjugate set of "rational points" containing $t=0$ and zero limits everywhere else.
Etc.