arXiv subjects
Publications and source records attributed to Denis Shatskov.
We proved that difference function $ψ_Θ-ψ_{Θ'}$ for almost all pairs $Θ$, $Θ'$ in cases $m=1$, $n=2$ or $m\geqslant2$ and $n=1$ changes its sign infinity many times as $t\rightarrow+\infty$.
We define two-dimensional Dirichlet spectrum (with respect to Euclidean norm) as D_2=λ\in\mathbf{R} | \exists \mathbf{v}=(v_1,v_2)\in \mathbf {R}^2: \limsup\limits_{t\rightarrow\infty} {t\cdotψ_{v}^2(t)}=λ, where ψ_{v}(t)=\min\limits_{1\leqslant q\leqslant t}\sqrt{|q v_1|^2+|q v_2|^2} is the two-dimensional "irrationality measure function". Our main result states the equality D_2=[0; 2/sqrt{3}].
We study asymptotics for the intergal of irrationality measure functions.