Let $\xi =[a_0;a_1,a_2,\cdots ,a_i,\cdots ]$ be an irrational number in simple continued fraction expansion, $p_i/q_i=[a_0;a_1,a_2,\cdots ,a_i]$, $M_i=q_i^2|\xi -p_i/q_i|$. In this note we find a function $G(R,r)$ such that $$\begin{array}{ccc}& M_{n+1}<R\text{and}{M}_{n-1}<r\text{imply}{M}_n>G(R,r),\hfill & \hfill M_{n+1}>R\text{and}{M}_{n-1}>r\text{imply}{M}_n<G(R,r).\end{array}$$ Together with a result the author obtained, this shows that to find two best approximation functions $\tilde{H}(R,r)$ and $\tilde{L}(R,r)$ is a well-posed problem. This problem has not been solved yet.

In this paper we characterize the closures of arithmetic progressions in the topology T on the set of positive integers with the base consisting of arithmetic progressions {an + b} such that if the prime number p is a factor of a, then it is also a factor of b. The topology T is called the common division topology.

A topological space $X$ is totally Brown if for each $n\in \mathbb{N}\setminus \left\{1\right\}$ and every nonempty open subsets $U_1,U_2,...,U_n$ of $X$ we have ${\mathrm{cl}}_X\left(U_1\right)\cap {\mathrm{cl}}_X\left(U_2\right)\cap \cdots \cap {\mathrm{cl}}_X\left(U_n\right)\ne \varnothing$. Totally Brown spaces are connected. In this paper we consider a topology ${\tau }_S$ on the set $\mathbb{N}$ of natural numbers. We then present properties of the topological space $(\mathbb{N},{\tau }_S)$, some of them involve the closure of a set with respect to this topology, while others describe subsets which are either totally Brown or totally separated. Our theorems generalize results proved by P. Szczuka in 2013, 2014, 2016 and by...

Canonical number systems in the ring of gaussian integers $\mathbb{Z}\left[i\right]$ are the natural generalization of ordinary $q$-adic number systems to $\mathbb{Z}\left[i\right]$. It turns out, that each gaussian integer has a unique representation with respect to the powers of a certain base number $b$. In this paper we investigate the sum of digits function ${\nu }_b$ of such number systems. First we prove a theorem on the sum of digits of numbers, that are not divisible by the $f$-th power of a prime. Furthermore, we establish an Erdös-Kac type theorem...

The article studies the cubic mapping graph $\Gamma \left(n\right)$ of ${\mathbb{Z}}_n\left[\mathrm{i}\right]$, the ring of Gaussian integers modulo $n$. For each positive integer $n>1$, the number of fixed points and the in-degree of the elements $\overline{1}$ and $\overline{0}$ in $\Gamma \left(n\right)$ are found. Moreover, complete characterizations in terms of $n$ are given in which ${\Gamma }_2\left(n\right)$ is semiregular, where ${\Gamma }_2\left(n\right)$ is induced by all the zero-divisors of ${\mathbb{Z}}_n\left[\mathrm{i}\right]$.