On an example by Poincar\'{e} and sums with Kronecker sequence

TL;DR

This paper analyzes Poincaré's example related to sums of continuous periodic functions with irrational shifts, providing a unified, multi-dimensional formulation, and discusses the non-existence of universal functions and new smoothness results involving Diophantine exponents.

Contribution

It offers a comprehensive, multi-dimensional generalization of recent results, introduces new smoothness insights involving Diophantine exponents, and clarifies the non-existence of universal continuous functions.

Findings

Unified multi-dimensional formulation of Poincaré's example

Proof of non-existence of universal continuous functions

New smoothness results involving Diophantine exponents

Abstract

This short and simple communication is motivated by recent papers by L. Colzani and A. Kochergin. We give a brief analysis of an example by Poincar\'{e} related to sums of the type $ \sum_{k=0}^{t-1} f(k{\alpha}+{x})$ where $f$ is a continuous periodic function and $\alpha$ is irrationaland its recent generalisations. Most of the constructions under consideration are well-known. In this note, we just wanted to bring all the results together and give a general and improved multi-dimensional formulation of a recent result by A. Kochergin, prove non-existence of a universal continuous function and discuss some of the related results in terms of Diophantine Approximation. In particular, in our opinion smoothness results involving Diophantine exponents $\omega$, $\hat{\omega}$ and $\hat{\lambda}$ had never been documented before.