Fabry gap theorem

Remarks

The condition $\lambda_k / k \to \infty$ says that the exponents that occur have density zero among the nonnegative integers. It is weaker than the condition $\lambda_{k+1} \ge q \lambda_k$ with $q > 1$ of the Hadamard gap theorem, so the Fabry theorem contains the Hadamard theorem. For example, the series $\sum_{k \ge 1} z^{k^2}$ has radius $1$, and $k^2 / k = k \to \infty$, so the unit circle is a natural boundary; the Hadamard theorem does not apply here, because $(k+1)^2 / k^2 \to 1$. The gap condition cannot simply be dropped: the series $\sum_{k \ge 0} z^k = 1/(1-z)$ has radius $1$ and continues analytically across every point of the unit circle except $z = 1$. The proof is harder than for the Hadamard theorem and is not sketched here.

Going deeper

For a student

Usually a power series has only a few singular points on its circle of convergence, and the function continues across the rest. Gap theorems show the opposite behaviour for lacunary series. The density condition on the exponents is the essential hypothesis, and it is much weaker than a fixed ratio between consecutive exponents. It shows that lacunary series are a simple source of functions with a natural boundary.