Asymptotically optimal packings of arithmetic progressions with prime differences

Abstract

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 $E\subseteq\mathbb{N}$, define $M_E(n)$ analogously using $B_e$, $e\in E$. Let $\mathcal{P}(x)=\{p\le x:p\text{ is prime}\}$. We prove $m_{\mathcal{P}(\sqrt n)}(n)=\left(\frac43+o(1)\right)\frac{n^{3/2}}{\ln n}$ and $M_{\mathcal{P}(n)}(n)=\left(\frac16+o(1)\right)\frac{n^3}{\ln n}$ as $n\to\infty$. These asymptotic formulas attain the known lower bounds and settle two conjectures of Alon, Dębski, Grytczuk and Przybyło concerning prime differences. The proof combines a cyclic phase-selection principle with lattice covering estimates and a decomposition into regular blocks of primes.