arXiv subjects
Publications and source records attributed to Nikolay G. Moshchevitin.
This paper is a survey of old and recent results related to Khintichine's singular matrices and their applications in the theory of Diophantine approximations. The paper is written in Russian. English version should appear in "Russian Mathematical Surveys" in the beginning of 2010.
Probably we have observed a new simple phenomena dealing with approximations to two real numbers.
We prove a result on the existence of linear forms of a given Diophantine type.
We give a sketch for an alternative proof of a recent result by J. Tseng.
We prove the existence of two-dimensional good lattice points in thick multiplicative subgroups modulo $p$.
We prove a new quantitative result on the degeneracy of the dimension of the subspace spanned by the best Diophantine approximations for a linear form.
For $α, β, δ\in [0,1], α+β= 1 $ we consider sets $$ {\rm BAD}^* (α, β;δ) = \left\{ξ= (ξ_1,ξ_2) \in [0,1]^2: ,\inf_{p\in \mathbb{N}} \max \{(p\log(p+1))^α||pξ_1||, (p\log (p+1))^β||pξ_2||\} \ge δ\right\}. $$ We prove that for different $(α_1,β_1), (α_2,β_2), α_1 +β_1 = α_2 +β_2 = 1 $ and $δ$ small enough $$ {\rm BAD}^* (α_1, β_1 ;δ) \bigcap {\rm BAD}^* (α_2, β_2 ;δ) \neq \varnothing . $$ Our result is based on A. Khintchine's construction and an original method due to Y. Peres and W. Schlag.
We prove that for any real polynomial $f(x) \in\mathbb{R} [x]$ the set $$ \{α\in \mathbb{R}: \liminf_{n\to \infty} n\log n ||αf(n)|| >0\} $$ has positive Hausdorff dimension. Here $||ξ||$ means the distance from $ξ$ to the nearest integer. Our result is based on an original method due to Y. Peres and W. Schlag.