Hadamard gap theorem

Definition. A sequence $\{n_k\}$ of nonnegative integers is said to be lacunary if there is some $\delta > 0$ such that $n_{k + 1}/n_k \geq 1 + \delta$ for all $k \geq 1$.

A typical example of a lacunary sequence is $n_k = 2^k$.

Theorem (Hadamard Gap Theorem).  Suppose $\{n_k\}$ is a lacunary sequence and the power series $f(z) = \sum_{n = 0}^\infty a_n z^n$ with $a_n$ nonzero precisely when $n = n_k$ for some $k$ and has radius of convergence 1. Then $f(z)$ cannot be analytically extended to any larger region containing a point on the unit circle.

Proof: Suppose $f(z)$ extends to an analytic function on a region containing the open unit disk $B$ and a point $w_0$ on the unit circle. WLOG, we may assume $w_0 = 1$. Then $f(z)$ is analytic on a region $\Omega$ containing $B \cup \{1\}$. Define $g(w) = f(w^m(1 + w)/2)$ where $m$ is a positive integer. Since $1^m(1 + 1)/2 = 1$ and $|w^m(1+w)/2| = |w|^m|1+w|/2 < |w|^m \leq 1$ when $|w| \leq 1$ and $w \neq 1$ because $|1 + w| < 2$, so $g(w)$ is well-defined when $|w| \leq 1$, hence when $|w| < 1 + \epsilon$ for some $\epsilon > 0$. Write $g(w) = \sum_{m = 0}^\infty b_m w^m$ for $|w| < 1 + \epsilon$. Note that the powers of $w$ in the terms of $[w^m(1 + w)/2]^{n_k}$ range from $n_k m$ to $n_k (m + 1)$. Since $\{n_k\}$ is lacunary, we can choose $m$ to be so large that $n_{k + 1} m > n_k (m + 1)$ for all $k$, then the powers of $w$ in all of $[w^m(1 + w)/2]^{n_k}$ are distinct. Now
\[ \sum_{n = 0}^{n_k} a_n z^n = \sum_{m = 0}^{n_k (m + 1)} b_m w^m. \]
When $|w| < 1 + \epsilon$, RHS converges as $k \rightarrow \infty$. But LHS is the partial sums of $\sum_{k = 1}^\infty a_{n_k} z^{n_k}$ as $k$ varies, so the power series converges for all $z = w^m(1 + w)/2$ where $|w| < 1 + \epsilon$, which includes the case $z = 1$. The image of $B(0, 1 + \epsilon)$ under the map $w^m(1 + w)/2$ is an region $W$ containing 1, so the power series of $f(z)$ at 0 converges on $W$, contradicting its radius of convergence being 1. QED.

Reference:
Gamelin - Complex Analysis
Rudin - Real and Complex Analysis (3rd Ed.)