Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define $r_N^{\left(m\right)}\left(\omega \right)$ to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that $r_N^{\left(m\right)}\left(\omega \right)<K$ for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.

In uniform distribution theory, discrepancy is a quantitative measure for the irregularity of distribution of a sequence modulo one. At the moment the concept of digital (t,s)-sequences as introduced by Niederreiter provides the most powerful constructions of s-dimensional sequences with low discrepancy. In one dimension, recently Faure proved exact formulas for different notions of discrepancy for the subclass of NUT digital (0,1)-sequences. It is the aim of this paper to generalize the concept...

In this paper we derive a sequence from a movement of center of~mass of arbitrary two planets in some solar system, where the planets circle on concentric circles in a same plane. A trajectory of center of mass of the planets is discussed. A sequence of points on the trajectory is chosen. Distances of the points to the origin are calculated and a distribution function of a sequence of the distances is found.

We present a method for constructing almost periodic sequences and functions with values in a metric space. Applying this method, we find almost periodic sequences and functions with prescribed values. Especially, for any totally bounded countable set $X$ in a metric space, it is proved the existence of an almost periodic sequence ${\left\{{\psi }_k\right\}}_{k\in \mathbb{Z}}$ such that $\{{\psi }_k;k\in \mathbb{Z}\}=X$ and ${\psi }_k={\psi }_{k+lq\left(k\right)}$, $l\in \mathbb{Z}$ for all $k$ and some $q\left(k\right)\in \mathbb{N}$ which depends on $k$.