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 $1254$.

Suppose that $G$ is a locally compact abelian group with a Haar measure $\mu$. The $\delta$-ball $B_{\delta }$ of a continuous translation invariant pseudo-metric is called $d$-dimensional if $\mu \left(B_{2{\delta }^{\prime }}\right)\le 2^d\mu \left(B_{{\delta }^{\prime }}\right)$ for all ${\delta }^{\prime }\subset (0,\delta ]$. We show that if $A$ is a compact symmetric neighborhood of the identity with $\mu \left(nA\right)\le n^d\mu \left(A\right)$ for all $n\ge dlogd$, then $A$ is contained in an $O\left(d{log}^{3}d\right)$-dimensional ball, $B$, of positive radius in some continuous translation invariant pseudo-metric and $\mu \left(B\right)\le exp\left(O\right(dlogd\left)\right)\mu \left(A\right)$.

We show that the number of squares in an arithmetic progression of length $N$ is at most $c_1{N}^{3/5}{\left(\log N\right)}^{c_2}$, for certain absolute positive constants $c_1$, $c_2$. This improves the previous result of Bombieri, Granville and Pintz [1], where one had the exponent $\frac{2}{3}$ in place of our $\frac{3}{5}$. The proof uses the same ideas as in [1], but introduces a substantial simplification by working only with elliptic curves rather than curves of genus $5$ as in [1].

A geometric progression of length k and integer ratio is a set of numbers of the form $a,ar,...,a{r}^{k-1}$ for some positive real number a and integer r ≥ 2. For each integer k ≥ 3, a greedy algorithm is used to construct a strictly decreasing sequence ${\left(a_i\right)}_{i=1}^{\infty }$ of positive real numbers with a₁ = 1 such that the set $G^{\left(k\right)}={\bigcup }_{i=1}^{\infty }(a_{2i},a_{2i-1}]$ contains no geometric progression of length k and integer ratio. Moreover, $G^{\left(k\right)}$ is a maximal subset of (0,1] that contains no geometric progression of length k and integer ratio. It is also proved that there is...

We consider the Golomb and the Kirch topologies in the set of natural numbers. Among other results, we show that while with the Kirch topology every arithmetic progression is aposyndetic, in the Golomb topology only for those arithmetic progressions $P(a,b)$ with the property that every prime number that divides $a$ also divides $b$, it follows that being connected, being Brown, being totally Brown, and being aposyndetic are all equivalent. This characterizes the arithmetic progressions which are aposyndetic...