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arXiv · 2310.05003

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

Nikolay Moshchevitin
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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.

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Nikolay Moshchevitin. 2023-10-08. On an example by Poincar\'{e} and sums with Kronecker sequence. https://arxiv.org/abs/2310.05003

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