What is known about $\displaystyle \sum_k{a^{b^k}}$? 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.
 A: 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.
A: 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.
