Behavior of lacunary series at the natural boundary

TL;DR

This paper develops a local theory for lacunary Dirichlet series near the natural boundary, revealing universal and self-similar singular behaviors, with applications in Fourier analysis, number theory, and dynamical systems.

Contribution

It introduces new asymptotic and Borel summability results for lacunary series with general coefficients, extending understanding of their boundary behavior and applications.

Findings

Derived blow-up rates along the imaginary axis.

Established Borel summability and exact local expansions.

Connected boundary behavior to Julia sets and Mandelbrot set structures.

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.