A family $\mathcal F$ of finite subsets of $\mathbb Z$ is packed into $[N]$ if suitable integer translates of its members are pairwise disjoint subsets of $[N]$. We study two prescribed-difference packing problems of Alon, D\k{e}bski, Grytczuk and Przyby{\l}o for the arithmetic progressions $A_d=\{d,2d,\ldots,\lfloor n...
For $k\geq2$, let $\alpha_k$ be the supremum of the exponents $a$ for which almost every integer $n$ has $k$ distinct divisors in a multiplicative interval of relative length $(\log n)^{-a}$. Select each positive integer $i$ independently with probability $1/i$, forming a random set $\mathbf A$, and let $\beta_k$ be th...
For a positive integer $n$, put $A_d=\{id:1\le i\le\lfloor n/d\rfloor\}$ for $1\le d\le n$ and $B_d=\{id:1\le i\le n\}$ for $d\in\mathbb{N}$. For $D\subseteq\{1,\ldots,n\}$, let $m_D(n)$ be the minimum length of an integer interval containing pairwise disjoint shifted copies of $A_d$ for all $d\in D$. For a finite set...
