Distribution of Kronecker sequences

Abstract

This short and simple communication is motivated by recent papers [1,5,15] as well as by the discussion on Poincaré's and Bohl's results in [35], Chapter 8. We give a brief analysis of an example constructed by Poincaré and 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 [15], 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 ω, ω and λ had never been documented before.

A Note on Badly Approximable Affine Forms and Winning Sets

“…In particular, this shows that there are singular matrices A with λ(Bad A ) = 0. The definition of the Diophantine class of matrices is stated in terms of the sequence of best approximations and is tightly related and motivated by the discussions in the papers [Mos23] and [Kim23]. We note that in [Kim23], another complementary Diophantine class is defined, which is a subclass of the singular matrices, and it is proved there that for such matrices A one has λ(Bad A ) = 1.…”

“…The proof of the following result is motivated by [Kim23, Theorem 1.7] and is based on an application of Minkowski successive minima theory and modifies the proof from [Mos23].…”

Сингулярные Диофантовы Системы А. Я. Хинчина И Их Применение

“…[101]) и Н. Г. Мощевити-ным в общем случае (см. [102], [103]). Ниже мы приведем формулировки этих результатов и дадим некоторые комментарии.…”

“…Доказательства теоремы 60 можно найти в работах автора [102] и [103]. Од-нако там она сформулирована с худшим значением показателя гладкости w(m).…”

“…Од-нако там она сформулирована с худшим значением показателя гладкости w(m). Тем не менее, доказательство из [102], [103] дословно проходит для показате-ля (102). К сожалению, соответствующая выкладка проведена только в дис-сертации автора [104].…”