Behavior of lacunary series at the natural boundary

Abstract

We develop a local theory of lacunary Dirichlet series of the form ∑ k = 1 ∞ c k exp ( − z g ( k ) ) , Re ( z ) > 0 as z approaches the boundary i R , under the assumption g ′ → ∞ 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 ∈ N , behavior near the boundary is roughly of the standard form Re ( z ) − b ′ Q ( x ) where Q ( x ) = 1 / q if x = p / q ∈ Q and zero otherwise. The Bötcher map at infinity of polynomial iterations of the form x n + 1 = λ P ( x n ) , | λ | < λ 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 , λ 0 = 1 , and the Julia set is the graph of this Fourier expansion in the main cardioid of the Mandelbrot set.