This paper is a continuation of a recent paper [2], in which the authors studied some Markov matrices arising from a mapping T:ℤ → ℤ, which generalizes the famous 3x+1 mapping of Collatz. We extended T to a mapping of the polyadic numbers $\widehat{\mathbb{Z}}$ and construct finitely many ergodic Borel measures on $\widehat{\mathbb{Z}}$ which heuristically explain the limiting frequencies in congruence classes, observed for integer trajectories.

Let $\sum _{n=1}^{\infty }{a}_n$ be a convergent series of positive real numbers. L. Olivier proved that if the sequence $\left(a_n\right)$ is non-increasing, then $\underset{n\to \infty }{lim}n{a}_n=0$. In the present paper: (a) We formulate and prove a necessary and sufficient condition for having $\underset{n\to \infty }{lim}n{a}_n=0$; Olivier’s theorem is a consequence of our Theorem . (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the $ℐ$-convergence, that is a convergence according to an ideal $ℐ$ of subsets of $\mathbb{N}$. Again, Olivier’s theorem is a consequence of...

This paper generalizes some results from another one, namely [3]. We have studied the issues of expressing natural numbers as a sum of powers of natural numbers in paper [3]. It means we have studied sets of type $A=\{n_1^{k_1}+n_2^{k_2}+\cdots +n_m^{k_m}\mid n_i\in \mathbb{N}\cup \left\{0\right\},i=1,2\cdots ,m,(n_1,n_2,\cdots ,n_m)\ne (0,0,\cdots ,0)\},$ where $k_1,k_2,\cdots ,k_m\in \mathbb{N}$ were given natural numbers. Now we are going to study a more general case, i.e. sets of natural numbers that are expressed as sum of integral parts of functional values of some special functions. It means that we are interested in sets of natural numbers in the form $$k=\left[f_1\left(n_1\right)\right]+\left[f_2\left(n_2\right)\right]+\cdots +\left[f_m\left(n_m\right)\right].$$

In this paper we study the set of statistical cluster points of sequences in $m$-dimensional spaces. We show that some properties of the set of statistical cluster points of the real number sequences remain in force for the sequences in $m$-dimensional spaces too. We also define a notion of $\Gamma$-statistical convergence. A sequence $x$ is $\Gamma$-statistically convergent to a set $C$ if $C$ is a minimal closed set such that for every $ϵ>0$ the set $\{k\rho (C,x_k)\ge ϵ\}$ has density zero. It is shown that every statistically bounded sequence...

Les ensembles “propres” pour une suite de Sidon sont caractérisés par une propriété de convergence des séries lacunaires à spectre dans la suite.