Spectral synthesis for exponentials and logarithmic length

TL;DR

This paper investigates the hereditary completeness of exponential systems with generating functions that are small outside lacunary intervals, establishing a criterion based on the logarithmic length of these intervals.

Contribution

It provides a new criterion linking hereditary completeness of exponential systems to the infinite logarithmic length of union of lacunary intervals, under certain conditions.

Findings

Hereditary completeness is characterized by infinite logarithmic length of intervals.

The study connects the size of generating functions outside lacunary sequences to completeness.

Technical conditions are identified for the equivalence to hold.

Abstract

We study hereditary completeness of systems of exponentials on an interval such that the corresponding generating function $G$ is small outside of a lacunary sequence of intervals $I_k$. We show that, under some technical conditions, an exponential system is hereditarily complete if and only if the logarithmic length of the union of these intervals is infinite, i.e., $\sum_k\int_{I_k} \frac{dx}{1+|x|}=\infty$.