High indices theorem

Remarks

In the language of Abel summation, the theorem says that a series with Hadamard gaps that is Abel summable is convergent. So the gap condition acts as a Tauberian condition, and no condition on the size of the coefficients $a_k$ is needed. The proof is considerably harder than that of Tauber's theorem. The gap hypothesis cannot be dropped: $\sum_{n \ge 0} (-1)^n x^n = 1/(1 + x) \to 1/2$ as $x \to 1^-$, but $\sum (-1)^n$ diverges. As a consequence, $f(x) = \sum_{k \ge 0} (-1)^k x^{2^k}$ has no limit as $x \to 1^-$, because $\sum_k (-1)^k$ diverges.

Going deeper

For a student

Abel's theorem goes from convergence to Abel summability, and Tauberian theorems go back under extra hypotheses. Usual Tauberian conditions bound the terms, as in $n a_n = O(1)$. Here the extra hypothesis is only on where the nonzero terms sit. The same gap condition appears in the Hadamard gap theorem, where it forces the unit circle to be a natural boundary.