Publications

Publications and preprints are listed from newest to oldest.

6. Bounded trajectories of quasi-rays on homogeneous spaces and Diophantine approximation with weight functions (2026)

Dmitry Kleinbock and Vasiliy Neckrasov

PreprintSubmitted

Abstract

Let

G

be a connected semisimple real Lie group,

\Gamma

an irreducible lattice in

G

and

X=G/\Gamma

. Let

F=\{g_t:t\ge 0\}

be a non-quasiunipotent one-parameter subsemigroup of

G

. Then it is known that the set of points in

X

with bounded

F

-trajectories has full Hausdorff dimension. In addition, if

U

is the expanding horospherical subgroup relative to

g_1

, then for any

x\in X

the set of points

u\in U

such that the

F

-trajectory of

ux

is bounded has full Hausdorff dimension. In this paper we take

U

to be a horospherical subgroup of

G

and apply Shi's equidistribution theorem for elements of the expanding cone with respect to

U

to describe a class of subsets

F

G

, not presupposing the group structure, for which the above full Hausdorff dimension statements also hold. As an application, we prove that the set of badly approximable matrices in the set-up of Diophantine approximations with quasimultiplicative weight functions has full Hausdorff dimension.

5. Uniform Diophantine approximation with restrictions via total density of collections of subspaces (2026)

Leo Hong, Dmitry Kleinbock, and Vasiliy Neckrasov

PreprintSubmitted

arXiv

Abstract

In 1926 Khintchine introduced a topological argument proving the existence of uncountably many nontrivial singular linear forms of

n\geq 2

variables. Throughout the years, this argument has been extensively modified and generalized. Most recently, Kleinbock et al. (2025) introduced a general framework of Diophantine systems and showed that a certain topological property called total density implies a far-reaching generalization of Khintchine's result. We describe a way to establish total density for a variety of Diophantine systems, and thus prove that the sets of singular objects are uncountable and dense in a wide range of set-ups in Diophantine approximation. As a special case, we establish such a result for inhomogeneous approximation, proving the existence of uncountably many singular systems of affine forms with a fixed translation part. One can also consider approximation with prime denominators, or more generally, approximation under some strong restrictions on numerators and denominators.

4. Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle (2026)

Vasiliy Neckrasov

Mathematika 72, e70089Published

arXiv Journal

Abstract

In 2019 Kleinbock and Wadleigh proved a “zero-one law” for uniform inhomogeneous Diophantine approximations. We generalize this statement to arbitrary weight functions and establish a new and simple proof of this statement, based on the transference principle. We also give a complete description of the sets of

g

-Dirichlet pairs with a fixed matrix in this set-up from the Lebesgue-measure point of view. As an application, we consider the set of badly approximable matrices and give a characterization of bad approximability in terms of inhomogeneous approximations. All the aforementioned metrical descriptions work, and can sometimes be strengthened, for weighted Diophantine approximations.

3. Metric theory of inhomogeneous Diophantine approximations with a fixed matrix (2025)

Nikolay Moshchevitin and Vasiliy Neckrasov

PreprintSubmitted

arXiv

Abstract

In this paper we develop a metric theory of inhomogeneous Diophantine approximation for the case of a fixed matrix. We use the transference principle to connect uniform Diophantine properties of a pair

(\Theta,\boldsymbol{\eta})

of a matrix and a vector with the asymptotic Diophantine properties of the transposed matrix

\Theta^{\top}

, and vice versa, the asymptotic Diophantine properties of a pair

(\Theta,\boldsymbol{\eta})

with asymptotic Diophantine properties of the transposed matrix. In these set-ups, we prove analogues of classical statements of metrical homogeneous Diophantine approximations and answer some open questions that were raised in recent works.

2. On Nontrivial Winning and Losing Parameters of Schmidt Games (2025)

Vasiliy Neckrasov and Eric Zhan

Results in Mathematics 80, 236Published

arXiv Journal

Abstract

In this paper we study the classical Schmidt game on two families of sets: one related to frequencies of digits in base-

2

expansions, and one connected to the set of the badly approximable numbers. Namely, we describe some nontrivial winning and losing parameters

(\alpha,\beta)

for these sets.

1. On some properties of irrational subspaces (2022)

Vasiliy Neckrasov

Uniform Distribution Theory 17 (1), 89–104Published

arXiv Journal

Abstract

In this paper we discuss some properties of completely irrational subspaces. We prove that there exist completely irrational subspaces that are badly approximable and, moreover, sets of such subspaces are winning in different senses. We get some bounds for Diophantine exponents of vectors that lie in badly approximable subspaces that are completely irrational; in particular, for any vector

\xi

from a two-dimensional badly approximable completely irrational subspace of

\mathbb{R}^d

one has

\widehat{\omega}(\xi)\leq\frac{\sqrt{5}-1}{2}

. Besides that, some statements about the dimension of subspaces generated by best approximations to a completely irrational subspace easily follow from properties that we discuss.