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Nikolay G. Moshchevitin

Publications and source records attributed to Nikolay G. Moshchevitin.

At least 19 recordsLinked to original sources
arXiv2021

A note on Dirichlet spectrum

We prove a result related to Dirichlet spectrum for simultaneous approximation to two real numbers in Euclidean norm and badly or very well approximability.

math.NT↗
arXiv2021

On transference principle and Nesterenko's linear independence criterion

We consider the problem of simultaneous approximation of real numbers $θ_1, \ldots,θ_n$ with rationals and the dual problem of approximating zero with the values of the linear form $x_0+θ_1x_1+\ldots+θ_nx_n$ at integer points. In this setting we analyse two transference inequalities obtained by Schmidt and Summerer. We present a rather simple geometric observation, which proves their result. We also derive several corollaries previously unknown. Particularly, we show that, together with the transference inequalities for uniform exponents, Schmidt and Summerer's inequalities imply the inequalities by Bugeaud and Laurent and "one half" of the inequalities by Marnat and Moshchevitin. Besides that, we show that our main construction provides a rather simple proof of Nesterenko's linear independence criterion.

math.NT↗
arXiv2019

On a modular form of Zaremba's conjecture

We prove that for any prime $p$ there is a divisible by $p$ number $q = O(p^{30})$ such that for a certain positive integer $a$ coprime with $q$ the ratio $a/q$ has bounded partial quotients. In the other direction we show that there is an absolute constant $C>0$ such that for any prime $p$ exist divisible by $p$ number $q = O(p^{C})$ and a number $a$, $a$ coprime with $q$ such that all partial quotients of the ratio $a/q$ are bounded by two.

math.NT↗
arXiv2018

Über die Funktionen des Irrationalitätsmaßes für zwei irrationalen Zahlen

For real $ξ$ we consider irrationality measure function $ψ_ξ(t) = \min_{1\le q \le t, \, q\in \mathbb{Z}} ||qξ||$. We prove that in the case $α\pm β\not\in \mathbb{Z}$ there exist arbitrary large values of $t$ with $|ψ_α(t) -ψ_β(t)| \ge\left(\sqrt{\frac{\sqrt{5}+1}{2}}-1\right) \min (ψ_α(t), ψ_β(t))$. This result is optimal.

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arXiv2016

Über die Funktionen des Irrationalitätsmaßes

We study two irrationality measure functions $ψ_α^{[2]} (t) $ and $ψ_α^{[2]*} (t)$ related to the "second best" approximations to a real numbers and prove some results on the structure of the corresponding Diophantine spectra. It happened that the first two elements of the spectrum for the function $ψ_α^{[2]*} (t)$ are associated with the numbers $\frac{1+\sqrt{5}}{2}$ and $e =\sum_{n=0}^\infty \frac{1}{n!}$.

math.NT↗
arXiv2012

Diophantine exponents for systems of linear forms in two variables

We improve on Jarn\'ık's inequality between uniform Diophantine exponent $α$ and ordinary Diophantine exponent $β$ for a system of $ n\ge 2$ real linear forms in two integer variables. Jarn\'ık (1949, 1954) proved that $β\ge α(α-1)$. In the present paper we give a better bound in the case $α>1$. We prove that β\ge 1/2(α^2-α+1+\sqrt{(α^2-α+1)^2 +4α^2(α-1)}) if 1\le α\le 2 1/2(α^2-1+\sqrt{(α^2-1)^2+4α(α-1)}) if α\ge 2

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arXiv2011

Positive integers: counterexample to W.M. Schmidt's conjecture

We show that there exist real numbers $α_1,α_2$ linearly independent over $\mathbb{Z}$ together with 1 such that for every non-zero integer vector $(m_1,m_2)$ with $m_1\ge 0$ and $m_2\ge 0$ one has $||m_1α_1+m_2α_2|| \ge 2^{-300} (\max(m_1, m_2))^{-σ}$ with $σ= 1.94696^+$.

math.NT↗
arXiv2011

On Kleinbock's Diophantine result

We give an elementary proof of a recent metrical Diophantine result by D. Kleinbock related to badly approximable vectors in affine subspaces.

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arXiv2010

Proof of W.M.Schmidt's conjecture concerning successive minima of a lattice

For a real $N\ge 1$ and a vector $ξ=(1,ξ_1,...,ξ_n)$ define a matrix $$ {\cal A} (ξ, N) = ({array}{ccccc} N^{-1} & 0& 0& ... &0 \cr N^{\frac{1}{n}} ξ_1 & -N^{\frac{1}{n}} & 0&... & 0 \cr N^{\frac{1}{n}} ξ_2 &0& -N^{\frac{1}{n}} & ... & 0 \cr ... &... &... &... \cr N^{\frac{1}{n}} ξ_n &0&0&... &- N^{\frac{1}{n}} {array}) $$ and a lattice $$ Λ(ξ, N) = {\cal A} (ξ, N)\mathbb{Z}^{n+1}. $$ Consider a convex 0-symmetric body $${\cal W} = \{z= (x,y_1,...,y_n)\in \mathbb{R}^{n+1}: \max (|x|, |y|)\le 1 \} >.$$ For a natural $l, 1\le l \le n+1$ let $μ_l (ξ, N)$ be the $l$-th successive minimum of ${\cal W}$ with respect to $ Λ(ξ, N)$. We prove that there exist real numbers $ξ_1,...,ξ_n$ linearly independent together with 1 over $\mathbb{Z}$, such that $μ_k (ξ, N) \to 0$ as $ N\to \infty$ and $μ_{k+2} (ξ, N) \to \infty$ as $ N\to \infty$.

math.NT↗
arXiv2010

Contribution to Vojtech Jarnik

We find new inequalities between uniform and individual Diophantine exponents for three-dimensional Diophantine approximations. Also we give a result for two linear forms in two variables. The results improves V.Jarnik's theorem (1954).

math.NT↗