John-Nirenberg inequality for Riemann type series

TL;DR

This paper improves the John-Nirenberg inequality for a class of Riemann-type series involving exponential functions with polynomial exponents, and provides a convergence criterion for these series.

Contribution

It offers a refined inequality specific to Riemann-type series with polynomial exponents and establishes conditions for their convergence.

Findings

Improved John-Nirenberg inequality for the series on specific intervals.

Established convergence criteria for the series.

Analyzed series behavior on intervals related to continued fraction convergents.

Abstract

We obtain an improvement of the John-Nirenberg inequality for the series of the form $\sum_{n=1}^{\infty}n^{-1}e^{2\pi i n^k x},\;k>2,$ on intervals consisting of points of a same convergent of their continued fractions. We also establish a convergence criterion for these series.