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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 $c_{V}\left(t\right)\neq0$. 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?

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?

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 $c_{V}\left(t\right)\neq0$. 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 $V$?

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 $c_{V}\left(t\right)\neq0$. 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?