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...