Hi!

I just came across the Ostroski-Hadamard gap theorem, and while I can understand the proofs as well as the principle that the series $\sum_{n=0}^\infty z^{2^n}$ ought to have a singularity at every $2^n$-th root of unity for every $n$, I feel I'm missing some intuition into what exactly is going on.

Specifically, there is certainly the intuition that the faster a power series' coefficients decrease, the larger the radius of convergence will be - say, comparing the geometric series with the exponential power series. When contrasted with lacunary series, this seems to fail: the coefficients seem to be increasingly "smaller", at least in an average sense, but the function becomes terribly ill-behaved. (One could try and argue that in the Cesàro sense the coefficients do tend to zero: if $\sum_{n=0}^\infty z^{2^n}=\sum_{k=0}^\infty a_k z^k$, then $\frac{1}{n}\sum_{k=0}^n a_k\approx\frac{\lfloor\log_2(n)\rfloor}{n}\rightarrow0$ as $n\rightarrow\infty$. On the other hand, the power series $\sum_{k=0}^\infty \frac{z^k}{k}$, while having the same radius of convergence, can easily, if non-uniquely, be analytically extended to the whole complex plane; I'd expect the same of any series of the form $\sum_{k=0}^\infty \frac{\log(k)}{k}z^k$.)

Can anyone share some insight?

Enumerative Combinatorics, vol..1, second ed., Exercise 4.46(c). $\endgroup$ – Richard Stanley Dec 2 '19 at 0:26