Let $a$ be a $0-1$ sequence with a finite number of terms equal to 1. The distance sequence ${\delta }^{\left(a\right)}$ of $a$ is defined as a sequence of the numbers of $(1-1)$-couples of given distances. The paper investigates such pairs of $0-1$ sequences $a,b$ that a is different from $b$ and ${\delta }^{\left(a\right)}={\delta }^{\left(b\right)}$.

Let the collection of arithmetic sequences ${\{d_{i}n+b_i:n\in \mathbb{Z}\}}_{i\in I}$ be a disjoint covering system of the integers. We prove that if $d_i=p^k{q}^l$ for some primes $p,q$ and integers $k,l\ge 0$, then there is a $j\ne i$ such that $d_i|d_j$. We conjecture that the divisibility result holds for all moduli.A disjoint covering system is called saturated if the sum of the reciprocals of the moduli is equal to $1$. The above conjecture holds for saturated systems with $d_i$ such that the product of its prime factors is at most $1254$.

A number of properties of a function which originally appeared in a problem proposed by Ramanujan are presented. Several equivalent representations of the function are derived. These can be used to evaluate the function. A new derivation of an expansion in inverse powers of the argument of the function is obtained, as well as rational expressions for higher order coefficients.

Let $𝒯$ be a system of disjoint subsets of ${\mathbb{N}}^{*}$. In this paper we examine the existence of an increasing sequence of natural numbers, $A$, that is an asymptotic basis of all infinite elements $T_j$ of $𝒯$ simultaneously, satisfying certain conditions on the rate of growth of the number of representations ${𝑟}_{𝑛}\left(𝐴\right);{𝑟}_{𝑛}\left(𝐴\right):=\left|\left\{({𝑎}_{𝑖},{𝑎}_{𝑗}):{𝑎}_{𝑖}\<{𝑎}_{𝑗};{𝑎}_{𝑖},{𝑎}_{𝑗}\in 𝐴;𝑛={𝑎}_{𝑖}+{𝑎}_{𝑗}\right\}\right|$, for all sufficiently large $n\in T_j$ and $j\in {\mathbb{N}}^{*}$ A theorem of P. Erdös is generalized.

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.