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Nikolay Moshchevitin

Publications and source records attributed to Nikolay Moshchevitin.

At least 19 recordsLinked to original sources
arXiv2026

Brjuno condition through best approximations and the linearization problem

We consider the classical analytic linearization problem for vector fields on the torus $\mathbb{T}^d$ close to a constant vector field $\omega$. Our goals are twofold. First, we provide a geometric framework in which the arithmetic condition governing analytic linearization arises naturally from the orbit of a unimodular lattice associated with $\omega$ under a diagonal flow on $\operatorname{SL}(d,\mathbb{Z})\backslash \operatorname{SL}(d,\mathbb{R})$. Within this framework, a summability condition emerges as the natural criterion for convergence. We prove that it is equivalent to several classical formulations of the Brjuno condition for linear forms, including those involving best approximation vectors and switching times of the diagonal flow. As a byproduct, we obtain a new quantitative linearization theorem with fully explicit estimates. In particular, the loss of analyticity of the conjugacy is controlled by a Brjuno function.

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arXiv2026

Weak approximations, Diophantine exponents and two-dimensional lattices

We study properties of Diophantine exponents of lattices and so-called related "weak" uniform approximations introduced in recent papers by Oleg German, in the simplest two-dimensional case. In contrast to the multidimensional case, in the two-dimensional case we can use a powerful tool of continued fractions. We develop an analog of Jarn\'ık's theory dealing with inequalities between the ordinary and uniform Diophantine exponents, which turned out to be related to mutual behaviour of irrationality measure functions for two real numbers.

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arXiv2025

On Diophantine exponents of lattices

We describe the spectrum of ordinary Diophantine exponents for $d$-dimensional lattices. The result reduces the problem to two-dimensional case and uses argument of metric theory.

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arXiv2025

Metric theory of inhomogeneous Diophantine approximations with a fixed matrix

In this paper we develop a metric theory of inhomogeneous Diophantine approximation for the case of a fixed matrix. We use transference principle to connect uniform Diophantine properties of a pair $(Θ, \pmbη)$ of a matrix and a vector with the asymptotic Diophantine properties of the transposed matrix $Θ^{\top}$, and vice versa, the asymptotic Diophantine properties of a pair $(Θ, \pmbη)$ with asymptotic Diophantine properties of the transposed matrix. In these setups, we prove analogues of classical statements of metrical homogeneous Diophantine approximations and answer some open questions that were raised in recent works.

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arXiv2025

A note on general isolation result in Diophantine Approximation

In the present paper we give very simple general statements which deal with approximation of a real number by rationals and are related to isolation phenomenon. In particular we study functions $ f(x)>f_1(x)>0$ such that existence of solutions $\frac{p}{q}$ of Diophantine inequality $ \left| α-\frac{p}{q}\right|< \frac{f(q)}{q^2} $ leads to the existence of solutions of inequality $ \left| α-\frac{p}{q}\right|< \frac{f_1(q)}{q^2} $.

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arXiv2025

Singularity, weighted uniform approximation, intersections and rates

A classical argument was introduced by Khintchine in 1926 in order to exhibit the existence of totally irrational singular linear forms in two variables. This argument was subsequently revisited and extended by many authors. For instance, in 1959 Jarnik used it to show that for $n \geq 2$ and for any non-increasing positive $f$ there are totally irrational matrices $A \in M_{m,n}({\mathbb R})$ such that for all large enough $t$ there are $\mathbf{p} \in {\mathbb Z}^m, \mathbf{q} \in {\mathbb Z}^n \smallsetminus \{0\}$ with $$\|\mathbf{q}\| \leq t \ \text{ and } \ \|A \mathbf{q} - \mathbf{p}\| \leq f(t).$$ We denote the collection of such matrices by $\mathrm{UA}^*_{m,n}(f)$. We adapt Khintchine's argument to show that the sets $\mathrm{UA}^*_{m,n}(f)$, and their weighted analogues $\mathrm{UA}^*_{m,n}(f, \mathbf{w})$, intersect many manifolds and fractals, and have strong intersection properties. For example, we show that: When $n \geq 2$, the set $\bigcap_{\mathbf{w}} \mathrm{UA}^*(f, \mathbf{w}) $, where the intersection is over all weights $\mathbf{w}$, is nonempty, and moreover intesects many manifolds and fractals; For $n \geq 2$, there are vectors in ${\mathbb R}^n$ which are simultaneously $k$-singular for every $k$, in the sense of Yu; when $n \geq 3$, $\mathrm{UA}^*_{1,n}(f) + \mathrm{UA}^*_{1,n}(f) = {\mathbb R}^n$. We also obtain new bounds on the rate of singularity which can be attained by column vectors in analytic submanifolds of dimension at least 2 in ${\mathbb R}^n$.

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arXiv2025

Bad approximability, bounded ratios and Diophantine exponents

For a real $m\times n$ matrix $\pmb{\xi}$, we consider its sequence of best Diophantine approximation vectors $ \pmb{x}_i \in \mathbb{Z}^n, \, i =1,2,3, ... $, the sequences of its norms $X_i = \|\pmb{x}_i\|$ and the norms of remainders $L_i = \|\pmb{\xi}\pmb{x}_i\|$. It is known that, in the cases $m=1$, bad approximability of $\pmb{\xi}$ is equivalent to the boundedness of ratios $\frac{X_{i+1}}{X_i}$, while for $n=1$ bad approximability of $\pmb{\xi}$ is equivalent to the boundedness of ratios $ \frac{L_i}{L_{i+1}}$. Moreover, carefully constructed example show that in the cases $m=1$ and $n=1$ boundedness of ratios $ \frac{L_i}{L_{i+1}}$ and $\frac{X_{i+1}}{X_i}$ respectively (the order of ratios changed), does not imply bad approximability of $\pmb{\xi}$. In the present paper, we study the impact of the boundedness of ratios on Diophantine properties of $\pmb{\xi}$, in particular, what restrictions it gives for Diophantine exponents $\omega(\pmb{\xi})$ and $\hat{\omega}(\pmb{\xi})$. One of our particular results deals with the case $m=n=2$. We prove that for $2\times 2 $ matrices $\pmb{\xi}$ boundedness of both ratios $ \frac{X_{i+1}}{X_i}, \frac{L_i}{L_{i+1}} $ implies inequality $\hat{\omega}(\pmb{\xi})\le \frac{4}{3}$ and that this result is optimal. Our methods combine parametric geometry of numbers as well as more classical tools.

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arXiv2024

On an example by Poincaré and sums with Kronecker sequence

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 Poincaré related to sums of the type $ \sum_{k=0}^{t-1} f(kα+{x})$ where $f$ is a continuous periodic function and $α$ 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 $ω$, $\hatω$ and $\hatλ$ had never been documented before.

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arXiv2024

Dirichlet improvability in $L_p$-norms

For a norm $F$ on $\mathbb{R}^2$, we consider the set of $F$-Dirichlet improvable numbers $\mathbf{DI}_F$. In the most important case of $F$ being an $L_p$-norm with $p=\infty$, which is a supremum norm, it is well-known that $\mathbf{DI}_F = \mathbf{BA}\cup \mathbb{Q}$, where $\mathbf{BA}$ is a set of badly approximable numbers. It is also known that $\mathbf{BA}$ and each $\mathbf{DI}_F$ are of measure zero and of full Hausdorff dimension. Using classification of critical lattices for unit balls in $L_p$, we provide a complete and effective characterization of $\mathbf{DI}_p:=\mathbf{DI}_{F^{[p]}}$ in terms of the occurrence of patterns in regular continued fraction expansions, where $F^{[p]}$ is an $L_p$-norm with $p\in[1,\infty)$. This yields several corollaries. In particular, we resolve two open questions by Kleinbock and Rao by showing that the set $\mathbf{DI}_{p}\setminus \mathbf{BA}$ is of full Hausdorff dimension, as well as proving some results about the size of the difference $\mathbf{DI}_{p_1}\setminus \mathbf{DI}_{p_2}$. To be precise, we show that the set difference of Dirichlet improvable numbers in Euclidean norm ($p=2$) minus Dirichlet improvable numbers in taxicab norm ($p=1$) and vice versa, that is $\mathbf{DI}_{2}\setminus \mathbf{DI}_{1}$ and $\mathbf{DI}_{1}\setminus \mathbf{DI}_{2}$, are of full Hausdorff dimension. We also find all values of $p$, for which the set $\mathbf{DI}_p^c\cap\mathbf{BA}$ has full Hausdorff dimension. Finally, our characterization result implies that the number $e$ satisfies $e\in \mathbf{DI}_p$ if and only if $p\in(1,2)\cup(p_0,\infty)$ for some special constant $p_0\approx2.57$.

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arXiv2024

Badly approximable grids and k-divergent lattices

For an m by n real matrix A, we investigate the set of badly approximable targets for A as a subset of the m-torus. It is well known that this set is large in the sense that it is dense and has full Hausdorff dimension. We investigate the relationship between its measure and Diophantine properties of A. On the one hand, we give the first examples of a non-singular matrix A such that the set of badly approximable targets has full measure with respect to some non-trivial algebraic measure on the torus. For this, we use transference theorems due to Jarnik and Khintchine, and the parametric geometry of numbers in the sense of Roy. On the other hand, we give a novel Diophantine condition on A that slightly strengthens non-singularity, and show that under the assumption that A satisfies this condition, the set of badly approximable targets is a null-set with respect to any non-trivial algebraic measure on the torus. For this we use naive homogeneous dynamics, harmonic analysis, and a novel concept we refer to as mixing convergence of measures.

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arXiv2024

On geometry of simultaneous approximation to three real numbers

Considering simultaneous approximation to three numbers, we study the geometry of the sequence of best approximations. We provide a sharper lower bound for the ratio between ordinary and uniform exponent of Diophantine approximation, optimal in terms of this geometry.

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arXiv2023

On Furstenberg's Diophantine result

We give a very simple and explicit exposition of the effective results on $\times a\times b$ by Bourgain, Lindenstrauss, Michel and Venkatesh.

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arXiv2022

On Korobov bound concerning Zaremba's conjecture

We prove in particular that for any sufficiently large prime $p$ there is $1\le a<p$ such that all partial quotients of $a/p$ are bounded by $O(\log p/\log \log p)$. For composite denominators a similar result is obtained. This improves the well--known Korobov bound concerning Zaremba's conjecture from the theory of continued fractions.

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