Let $V$ be an arbitrary set of (countably) 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 and is non-zero. Is $T_{V}$ necessarily finite (that is, would an "overconcentration" of radii on which $\varsigma_{V}\left(z\right)$ grows like $\frac{1}{1-\left|z\right|}$ as $z$ tends radially to the unit circle result in a contradiction against the boundedness of $\varsigma_{V}\left(z\right)$'s power series coefficients)? Or do there exist $V$ for which $T_{V}$ is infinite?

Part of the problem is that there is so much literature about the boundary behavior and singularities of power series that trying to find information about something this specific is like looking for the needle in the proverbial haystack. Any assistance would be much appreciated.

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 and is non-zero. Is $T_{V}$ necessarily finite (that is, would an "overconcentration" of radii on which $\varsigma_{V}\left(z\right)$ grows like $\frac{1}{1-\left|z\right|}$ as $z$ tends radially to the unit circle result in a contradiction against the boundedness of $\varsigma_{V}\left(z\right)$'s power series coefficients)? Or do there exist $V$ for which $T_{V}$ is infinite?

Part of the problem is that there is so much literature about the boundary behavior and singularities of power series that trying to find information about something this specific is like looking for the needle in the proverbial haystack. Any assistance would be much appreciated.

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 and is non-zero. Is $T_{V}$ necessarily finite (that is, would an "overconcentration" of radii on which $\varsigma_{V}\left(z\right)$ grows like $\frac{1}{1-\left|z\right|}$ as $z$ tends radially to the unit circle result in a contradiction against the boundedness of $\varsigma_{V}\left(z\right)$'s power series coefficients)? Or do there exist $V$ for which $T_{V}$ is infinite?

Part of the problem is that there is so much literature about the boundary behavior and singularities of power series that trying to find information about something this specific is like looking for the needle in the proverbial haystack. Any assistance would be much appreciated.

Let $V$ be an arbitrary set of (countably) 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 and is non-zero. Is $T_{V}$ necessarily finite (that is, would an "overconcentration" of radii on which $\varsigma_{V}\left(z\right)$ grows like $\frac{1}{1-\left|z\right|}$ as $z$ tends radially to the unit circle result in a contradiction against the boundedness of $\varsigma_{V}\left(z\right)$'s power series coefficients)? Or do there exist $V$ for which $T_{V}$ is infinite?

Part of the problem is that there is so much literature about the boundary behavior and singularities of power series that trying to find information about something this specific is like looking for the needle in the proverbial haystack. Any assistance would be much appreciated.