Hi!


I just came across the [Ostroski-Hadamard gap theorem](http://en.wikipedia.org/wiki/Ostrowski-Hadamard_gap_theorem), and while I can understand the proofs as well as the [principle](http://en.wikipedia.org/wiki/Lacunary_function#A_simple_example) 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?



Why are lacunary series so badly behaved?
You can read more about this in the excellent survey



> 
> J.-P. Kahane: A century of interplay between Taylor series, Fourier series and Brownian motion, Bull. London Math. Soc. **29**(1997), 257-279
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In particular you can learn from this survey that the phenomenon you mentioned is rather typical. It's definitely worth having a look at it. 


