Abstract:
Let $\alpha$ be an irrational number. For $n \in \mathbb{N}$, we consider sets of points $\alpha_j - j \alpha\, (\operatorname{mod} 1) - \beta_{k(j)}$, $j-0,\dots ,n$. Points $\beta_k$ divide the interval $[0,1]$ on $n+1$ segments $[\beta_k,\beta_{k+1}]$ which lengths are $\delta_k - \beta_{k+1} - \beta_k$. In the paper, with the help of continued fractions theory, we investigate the relation of indexes $j$ and $k(j)-k(j,n)$ of numbers $\alpha_j - \beta_{k(j)}$. For this purpose we introduce graphs of the left and of the right predecessors of a point $\alpha_j$. We calculated statistics of length $\delta_k - \delta_k(n+1)$. The last ones are compared with statistics of the distribution of lengths of analogous segments in the case when all of numbers $\alpha_j$ are uniformly distributed independent random values. The means in two statistics considered coincide, but the ratio of its dispersions may take arbitrary small value and arbitrary big value as well.
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