Behavior of lacunary series at the natural boundary

Abstract

We develop a local theory of lacunary Dirichlet series of the form $\sum\limits_{k=1}^{\infty}c_k\exp(-zg(k)), \Re(z)>0$ as $z$ approaches the boundary $i\RR$, under the assumption $g'\to\infty$ and further assumptions on $c_k$. These series occur in many applications in Fourier analysis, infinite order differential operators, number theory and holomorphic dynamics among others. For relatively general series with $c_k=1$, the case we primarily focus on, we obtain blow up rates in measure along the imaginary line and asymptotic information at $z=0$. When sufficient analyticity information on $g$ exists, we obtain Borel summable expansions at points on the boundary, giving exact local description. Borel summability of the expansions provides property-preserving extensions beyond the barrier. The singular behavior has remarkable universality and self-similarity features. If $g(k)=k^b$, $c_k=1$, $b=n$ or $b=(n+1)/n$, $n\in\NN$, behavior near the boundary is roughly of the standard form $\Re(z)^{-b'}Q(x)$ where $Q(x)=1/q$ if $x=p/q\in\QQ$ and zero otherwise. The B\"otcher map at infinity of polynomial iterations of the form $x_{n+1}=\lambda P(x_n)$, $|\lambda|<\lambda_0(P)$, turns out to have uniformly convergent Fourier expansions in terms of simple lacunary series. For the quadratic map $P(x) =x-x^2$, $\lambda_0=1$, and the Julia set is the graph of this Fourier expansion in the main cardioid of the Mandelbrot set.