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Nikolay Moshchevitin

Publications and source records attributed to Nikolay Moshchevitin.

At least 37 records · Page 2Linked to original sources
arXiv2021

On badly approximable numbers

Motivated by a wonderful paper by Ngoc Ai Van Nguyen, Anthony Poëls and Damien Roy, where a powerful method was introduced, we prove a criterion for a vector $\pmbα\in \mathbb{R}^d$ to be a badly approximable vector. Moreover we construct certain examples which show that a more general version of our criterion is not valid.

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arXiv2020

Singular vectors on manifolds and fractals

We generalize Khintchine's method of constructing totally irrational singular vectors and linear forms. The main result of the paper shows existence of totally irrational vectors and linear forms with large uniform Diophantine exponents on certain subsets of $\mathbb{R}^n$, in particular on any analytic submanifold of $\mathbb{R}^n$ of dimension $\ge 2$ which is not contained in a proper rational affine subspace.

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arXiv2020

An optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation

We provide a lower bound for the ratio between the ordinary and uniform exponent of both simultaneous Diophantine approximation and Diophantine approximation by linear forms in any dimension. This lower bound was conjectured by Schmidt and Summerer and already shown in dimension $2$ and $3$. This lower bound is reached at regular systems presented in the context of parametric geometry of numbers, and thus optimal.

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arXiv2019

Über die Winkel zwischen Unterräumen

We prove a metric statement about approximation of a $n$-dimensional linear subspace $A$ in $\mathbb{R}^d$ by $n$-dimensional rational subspaces. We consider the problem of finding a rational subspace $B$ of bounded height $H=H(B)$ for which the angle of inclination $ψ(A,B) $ is small in terms of $H$. In the simplest case $d=4, n=2$ we give a partial solution of a problem formulated by W.M. Schmidt in 1967.

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arXiv2018

Simultaneous Diophantine approximation: sums of squares and homogeneous polynomials

Let $f$ be a homogeneous polynomial with rational coefficients in $d$ variables. We prove several results concerning uniform simultaneous approximation to points on the graph of $f$, as well as on the hypersurface $\{f(x_1,\dots,x_d) = 1\}$. The results are first stated for the case $f(x_1,\dots,x_d) = x_1^2+\dots+x_d^2,$ which is of particular interest.

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arXiv2018

Popular Products and Continued Fractions

We prove bounds for the popularity of products of sets with weak additive structure, and use these bounds to prove results about continued fractions. Namely, we obtain a nearly sharp upper bound for the cardinality of Zaremba's set modulo $p$.

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arXiv2018

On an effective variation of Kronecker's approximation theorem avoiding algebraic sets

Let $Λ\subset \mathbb R^n$ be an algebraic lattice, coming from a projective module over the ring of integers of a number field $K$. Let $\mathcal Z \subset \mathbb R^n$ be the zero locus of a finite collection of polynomials such that $Λ\nsubseteq \mathcal Z$ or a finite union of proper full-rank sublattices of $Λ$. Let $K_1$ be the number field generated over $K$ by coordinates of vectors in $Λ$, and let $L_1,\dots,L_t$ be linear forms in $n$ variables with algebraic coefficients satisfying an appropriate linear independence condition over $K_1$. For each $\varepsilon > 0$ and $\boldsymbol a \in \mathbb R^n$, we prove the existence of a vector $\boldsymbol x \in Λ\setminus \mathcal Z$ of explicitly bounded sup-norm such that $$\| L_i(\boldsymbol x) - a_i \| < \varepsilon$$ for each $1 \leq i \leq t$, where $\|\ \|$ stands for the distance to the nearest integer. The bound on sup-norm of $\boldsymbol x$ depends on $\varepsilon$, as well as on $Λ$, $K$, $\mathcal Z$ and heights of linear forms. This presents a generalization of Kronecker's approximation theorem, establishing an effective result on density of the image of $Λ\setminus \mathcal Z$ under the linear forms $L_1,\dots,L_t$ in the $t$-torus~$\mathbb R^t/\mathbb Z^t$.

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arXiv2017

A note on badly approximable linear forms on manifolds

This paper is motivated by Davenport's problem and the subsequent work regarding badly approximable points in submanifolds of a Euclidian space. We study the problem in the area of twisted Diophantine approximation and present two different approaches. The first approach shows that, under a certain restriction, any countable intersection of the sets of weighted badly approximable points on any non-degenerate C^1 submanifold of R^n has full dimension. In the second approach we introduce the property of isotropically winning and show that the sets of weighted badly approximable points are isotropically winning under the same restriction as above.

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arXiv2016

Badly approximable points in twisted Diophantine approximation and Hausdorff dimension

For any j_1,...,j_n>0 with j_1+...+j_n=1 and any x \in R^n, we consider the set of points y \in R^n for which max_{1\leq i\leq n}(||qx_i-y_i||^{1/j_i})>c/q for some positive constant c=c(y) and all q\in N. These sets are the `twisted' inhomogeneous analogue of Bad(j_1,...,j_n) in the theory of simultaneous Diophantine approximation. It has been shown that they have full Hausdorff dimension in the non-weighted setting, i.e provided that j_i=1/n, and in the weighted setting when x is chosen from Bad(j_1,...,j_n). We generalise these results proving the full Hausdorff dimension in the weighted setting without any condition on x.

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arXiv2015

A note on weighted badly approximable linear forms

We prove a result in the area of twisted Diophantine approximation related to the theory of Schmidt games. In particular, under certain restrictions we give a affirmative answer to the analogue in this setting of a famous conjecture of Schmidt from Diophantine approximation.

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arXiv2014

Über die rationalen Punkte auf der Sphäre

We give a simple proof of a recent result by Kleinbock and Merrill concerning intrinsic approximations on sphere, in the simplest case of two-dimensional sphere in $\mathbb{R}^3$.

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arXiv2013

On Diophantine exponents in dimension 4

We obtain some new inequalities between the ordinary and the uniform Diophantine exponents for simultaneous Diophantine approximation to four real numbers.

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