Why are lacunary series so badly behaved? - MathOverflow

Why are lacunary series so badly behaved?

2012-03-04T20:00:30Z

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?

Answer by GH for Why are lacunary series so badly behaved?

2012-03-04T20:51:51Z

The mentioned gap theorem was generalized by Fabry (Acta Math. 1899, pp. 65-87): if the power series $f(z)=\sum_n a_n z^{\lambda_n}$ has radius of convergence $1$, and the exponents $\lambda_n\in\mathbb{N}$ satisfy $\lambda_n/n\to\infty$, then the unit circle is a natural boundary for $f(z)$.

Turán (Acta Math. Hung. 1947, pp. 21-29) gave a simple proof which might provide some insight into the phenomenon. His main inequality, from which he deduces the result, reads as follows:

$$ \max_{0\leq x\leq 2\pi}\ \left| \sum_{n=1}^N a_n e^{i\lambda_n x} \right| \leq \left(\frac{48\pi}{\delta}\right)^N \max_{a\leq x\leq a+\delta}\ \left| \sum_{n=1}^N a_n e^{i\lambda_n x} \right| $$

In other words, the key feature seams to be that on every arc of the unit circle, the partial sums are considerably bounded away from zero. For more details I would recommend to study Turán's paper.

Answer by Liviu Nicolaescu for Why are lacunary series so badly behaved?

2012-03-05T13:36:51Z

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

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.

Answer by Gerald Edgar for Why are lacunary series so badly behaved?

2012-03-05T14:37:23Z

Maybe your question is backwards. Natural boundary at the radius of convergence is the usual thing, and analytic continuation outside the circle of convergence is the fluke. Only VERY SPECIAL series have continuations.

Answer by Pietro Majer for Why are lacunary series so badly behaved?

2012-03-05T18:53:08Z

"Objection, the question assumes facts not in evidence!"

Talking about the general question as in the title, I wonder in what measure can we say that lacunary series are particularly badly behaved. Maybe the point is just that a lacunary form makes it easier to construct badly behaved series, which is slightly different. An example: we know that a real entire function $f$, say with real coefficients, may grow as fast as any given increasing function on $g:\mathbb{R}\to\mathbb{R}$, and building an example is easy by means of lacunary series. But $f(z+1)$ grows even faster, although the translation destroys the lacunary form.