On the "pits effect" of Littlewood and Offord

TL;DR

This paper investigates the 'pits effect' in entire functions with specific coefficient arguments, showing that certain conditions lead to exponential functions or regular growth with constant indicator.

Contribution

It characterizes the 'pits effect' for entire functions with coefficients having quadratic phase arguments, extending understanding of their growth and structure.

Findings

Functions with coefficients of form 2pi n^2a exhibit the pits effect.

Such functions are of completely regular growth with constant indicator.

Exponential decay on a ray implies the function is exponential.

Abstract

Suppose that the moduli of the coefficients of a power series are 1/n!, while the arguments are arbitrary. If an entire function f represented by such power series decreases exponentially on some ray, then it has to be an exponential. If the arguments of the coefficients are of the form 2pi n^2a, where a is irrational, then the function displays the so-called "pits effect". More precisely, under this condition, f is of completely regular growth with constant indicator.