arXiv subjects
Publications and source records attributed to Nikolay Moshchevitin.
We prove a quantitative version of the following statement: the unipotent flow orbit of a typical lattice in $\rm{SL}_2(\mathbb{R})/\rm{SL}_2(\mathbb{Z})$ is dense. Our quantitative result uses A. Weil's bounds for Kloostermann sums.
Consider irrational affine subspace $ A\subset \mathbb{R}^d$ of dimension $a$. We prove that the set $$ \{ξ=(ξ_1,...,ξ_d) \in {A}:\,\,\, \ q^{1/a}\cdot \max_{1\le i \le d} ||qξ_i|| \to \infty,\,\,\,\, q\to \infty\} $$ is an $α$-winning set for every $α\in (0,1/2]$
We give several results related to inhomogeneous approximations to two real numbers and badly approximable numbers. Our results are related to classical theorems by A. Khintchine (1926) and to an original method invented by Y. Peres and W. Schlag (2001).
Let $Θ= (θ_1,θ_2,θ_3)\in \mathbb{R}^3$. Suppose that $1,θ_1,θ_2,θ_3$ are linearly independent over $\mathbb{Z}$. For Diophantine exponents $$ α(Θ) = \sup \{γ>0:\,\,\, \limsup_{t\to +\infty} t^γψ_Θ(t) <+\infty \} ,$$ $$β(Θ) = \sup \{γ>0:\,\,\, \liminf_{t\to +\infty} t^γψ_Θ(t) <+\infty\} $$ we prove $$ β(Θ) \ge {1/2} ({α(Θ)}/{1-α(Θ)} +\sqrt{{α(Θ)}/{1-α(Θ)})^2 +{4α(Θ)}/{1-α(Θ)}}) α(Θ) $$
We prove that there exist arbitrarily small positive real numbers $ε$ such that every integral power $(1 + \vepsilon)^n$ is at a distance greater than $2^{-17} ε|\log \vepsilon|^{-1}$ to the set of rational integers. This is sharp up to the factor $2^{-17} |\log ε|^{-1}$. We also establish that the set of real numbers $α> 1$ such that the sequence of fractional parts $(\{α^n\})_{n \ge 1}$ is not dense modulo 1 has full Hausdorff dimension.
We establish that the set of pairs $(α, β)$ of real numbers such that $$ \liminf_{q \to + \infty} q \cdot (\log q)^2 \cdot \Vert q α\Vert \cdot \Vert q β\Vert > 0, $$ where $\Vert \cdot \Vert$ denotes the distance to the nearest integer, has full Hausdorff dimension in $\R^2$. Our proof rests on a method introduced by Peres and Schlag, that we further apply to various Littlewood-type problems
By means of Peres-Schlag's method we prove the existence of real numbers $α, β$ such that $$ \liminf_{q\to \infty} (q\log^2 q)||αq|| ||βq|| > 0.