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What is known about $\displaystyle \sum_k{a^{b^k}}$? I am very interested in the possible applications of this series.

I have asked about this on Mathematics Stack Exchange here.

I'm wondering if there has been any research on this particular type of series, or what similar ideas/topics are out there.

Please be gentle. I'm not an expert, but I'm wondering if someone could possibly point me in the right direction.

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The series $f(z)=\sum_{k=0}^{\infty}z^{b^k}$ is the most important example of a lacunary Taylor series, which have been studied quite a lot. One important theorem about them is the Hadamard gap theorem: $f$ does not extend analytically to any part of the boundary of the unit disc. This paper by Jensen, Pommerenke and Ramírez studies the properties of $f$ when $b=2$ (but many of the theorems probably generalize).

In the paper there are many theorems about how $f$ maps the radii $[0,\zeta] ,\zeta\in \partial \mathbb{D}$ of the unit disk. We for example have a ''central limit theorem'' from Salem and Zygmund: for any measurable $E\subset \partial \mathbb{D}$ we have $$m(\{\zeta \in E:\Re(f(1-2^{-t}\zeta))\leq x\sqrt{\frac{t}{2}})\to \frac{m(E)}{\sqrt{2\pi}}\int_{-\infty}^x e^{-\frac{y^2}{2}}dy$$ as $t\to \infty$, and the same holds for imaginary parts. Therefore, when we are ''close'' to the boundary of the disk, the bahavior of $f$ as a function of the direction $\zeta$ resembles normal distribution.

There is also an elegant law of iterated logarithms by Erdös, Gal and Weiss: $$\limsup_{t\to \infty}\frac{f((1-2^{-t})\zeta)}{\sqrt{t\log \log t}}=1$$ for almost every $\zeta \in \partial \mathbb{D}$.

According to the paper, $f$ also has the property that the image of the unit disc is the whole complex plane.

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The famous Weierstrass continuous nowhere differentiable function is $\Re\sum c^ne^{2\pi b^nx}.$ Setting $c=1$, we obtain your series with $a=e^{2\pi x}$.

Weierstrass function was studied in great detail by Hardy, TAMS 17 (1916) 301-325, and by many other authors.

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  • $\begingroup$ Isn't the Weierstrass function a sum of trigonometric exponentials $e^{2\pi i b^n x}$ with $|c|<1$? $\endgroup$ – Joni Teräväinen May 18 '14 at 10:15
  • $\begingroup$ Yes. Except that $c$ is missing in your formula. I just noticed that if you put $c=1$ you obtain your function. You did not ask any specific question, so I do not discuss the real meaning of putting $c=1$. $\endgroup$ – Alexandre Eremenko May 18 '14 at 12:50
  • $\begingroup$ I meant that the series $\sum_n c^n e^{2\pi i b^n}$ is the Weierstrass function. If you take $i$ away from the exponent, you get an analytic function instead of a nondifferentiable one. $\endgroup$ – Joni Teräväinen May 18 '14 at 18:54

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