Portrait of Vasiliy Neckrasov

Vasiliy Neckrasov

Diophantine approximations, homogeneous dynamics

PhD studentDepartment of Mathematics · Brandeis UniversityWaltham, Massachusetts

vneckrasov@gmail.comvneckrasov@brandeis.eduCVGoogle ScholarORCIDarXiv

Publications

Publications and preprints from newest to oldest.

7. Winning property of counterexamples to Uniform Littlewood's Conjecture (2026)arXiv

Vasiliy Neckrasov, Chengyang Wu, and Bohan YangPreprint

Abstract
In this paper, we prove that the set of counterexamples to the uniform Littlewood's conjecture proposed in \cite{BFK25}, that is, the set of pairs of real numbers (x,y)(x,y) satisfying lim sup⁡Q→+∞Q⋅min⁡1≤q≤Q⟨qx⟩⟨qy⟩>0, \limsup_{Q\to +\infty}Q\cdot \min_{1\leq q\leq Q}\langle qx\rangle\langle qy\rangle>0, is hyperplane absolute winning. We show that a stronger statement holds: the set of real pairs (x,y)(x,y) satisfying lim inf⁡m→+∞Qm⋅min⁡1≤q≤Qm⟨qx⟩⟨qy⟩>0, \liminf_{m\to +\infty}Q_m\cdot \min_{1\leq q\leq Q_m}\langle qx\rangle\langle qy\rangle>0, is hyperplane absolute winning if Qm+1≫QmτQ_{m+1} \gg Q_m^{\tau} for some τ>1\tau > 1. In particular, the above sets have full Hausdorff dimension in R2\mathbb{R}^2. In addition, we prove that these sets are absolute winning on every regular C2C^2 planar curve whose set of points of nonzero curvature is itself absolute winning on the curve. We also establish analogous results for certain lines.

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

Dmitry Kleinbock and Vasiliy NeckrasovPreprint

Abstract
Let GG be a connected semisimple real Lie group, Γ\Gamma an irreducible lattice in GG and X=G/ΓX=G/\Gamma. Let F={gt:t≥0}F=\{g_t:t\ge 0\} be a non-quasiunipotent one-parameter subsemigroup of GG. Then it is known that the set of points in XX with bounded FF-trajectories has full Hausdorff dimension. In addition, if UU is the expanding horospherical subgroup relative to g1g_1, then for any x∈Xx\in X the set of points u∈Uu\in U such that the FF-trajectory of uxux is bounded has full Hausdorff dimension. In this paper we take UU to be a horospherical subgroup of GG and apply Shi's equidistribution theorem for elements of the expanding cone with respect to UU to describe a class of subsets FF in GG, 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)arXiv

Leo Hong, Dmitry Kleinbock, and Vasiliy NeckrasovPreprint

Abstract
In 1926 Khintchine introduced a topological argument proving the existence of uncountably many nontrivial singular linear forms of n≥2n\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)arXivJournal

Vasiliy NeckrasovMathematika 72, e70089

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 gg-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)arXiv

Nikolay Moshchevitin and Vasiliy NeckrasovPreprint

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 (Θ,η)(\Theta,\boldsymbol{\eta}) of a matrix and a vector with the asymptotic Diophantine properties of the transposed matrix Θ⊤\Theta^{\top}, and vice versa, the asymptotic Diophantine properties of a pair (Θ,η)(\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)arXivJournal

Vasiliy Neckrasov and Eric ZhanResults in Mathematics 80, 236

Abstract
In this paper we study the classical Schmidt game on two families of sets: one related to frequencies of digits in base-22 expansions, and one connected to the set of the badly approximable numbers. Namely, we describe some nontrivial winning and losing parameters (α,β)(\alpha,\beta) for these sets.

1. On some properties of irrational subspaces (2022)arXivJournal

Vasiliy NeckrasovUniform Distribution Theory 17 (1), 89–104

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 ξ\xi from a two-dimensional badly approximable completely irrational subspace of Rd\mathbb{R}^d one has ω^(ξ)≤5−12\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.