Some Trigonometric Polynomials with Extremely Small Uniform Norm

TL;DR

This paper presents a specific example of trigonometric polynomials with very small uniform norms, highlighting the limitations of extending Sidon's inequality for lacunary polynomials.

Contribution

It provides a novel example illustrating the potential boundaries of Sidon's inequality in the context of lacunary trigonometric polynomials.

Findings

Demonstrates the existence of trigonometric polynomials with extremely small uniform norms.

Highlights limitations in extending Sidon's inequality.

Provides insights into the behavior of lacunary polynomials.

Abstract

An example of trigonometric polynomials with extremely small uniform norm is given. This example demonstrates the potential limits for extension of Sidon's inequality for lacunary polynomials in a certain direction.