发表机构
Center for Discrete Mathematics, Fuzhou University（福州大学离散数学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明素数差等差数列打包问题的两个渐近最优公式，达到已知下界并解决相关猜想，方法结合循环相位选择、格覆盖与素数规则块分解。
AI 中文摘要
对于正整数 $n$，当 $1\le d\le n$ 时令 $A_d=\{id:1\le i\le\lfloor n/d\rfloor\}$，当 $d\in\mathbb{N}$ 时令 $B_d=\{id:1\le i\le n\}$。对于 $D\subseteq\{1,\ldots,n\}$，设 $m_D(n)$ 为包含所有 $d\in D$ 的 $A_d$ 的两两不相交平移副本的最短整数区间长度。对于有限集 $E\subseteq\mathbb{N}$，类似地用 $B_e$（$e\in E$）定义 $M_E(n)$。令 $\mathcal{P}(x)=\{p\le x:p\text{ 为素数}\}$。我们证明当 $n\to\infty$ 时，$m_{\mathcal{P}(\sqrt n)}(n)=\left(\frac43+o(1)\right)\frac{n^{3/2}}{\ln n}$ 且 $M_{\mathcal{P}(n)}(n)=\left(\frac16+o(1)\right)\frac{n^3}{\ln n}$。这些渐近公式达到了已知下界，并解决了 Alon、Dębski、Grytczuk 和 Przybyło 关于素数差的两个猜想。证明结合了循环相位选择原理、格覆盖估计以及将素数分解为规则块的方法。
英文摘要
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, Dębski, Grytczuk and Przybyło for the arithmetic progressions $A_d=\{d,2d,\ldots,\lfloor n/d\rfloor d\}$ and $B_d=\{d,2d,\ldots,nd\}$. Let $m(n)$ and $M(n)$ denote the corresponding minimum packing lengths, with subscripts indicating restrictions on the admissible differences, and let $\mathcal P(x)$ denote the set of primes at most $x$. The key ingredient is a residue-class selection scheme that encodes pairwise intersection constraints by cyclic intervals. A lattice-covering argument yields a simultaneous admissible choice, permitting substantial overlap of the containing intervals and providing the sharp upper bounds needed for the bounded-diameter family and the prime-difference equal-cardinality family. For the bounded-diameter family, we prove $m(n)\sim m_{\mathcal P(\sqrt n)}(n)\sim 4n^{3/2}/(3\log n)$. For the equal-cardinality family with prime differences, we prove $M_{\mathcal P(n)}(n)\sim n^3/(6\log n)$. For the full equal-cardinality family, we show $M(n)\ge \left(\frac{19}{108}-o(1)\right)n^3/\log n$. Together with corresponding estimates for restricted ranges of differences, these results prove several conjectures of Alon--Dębski--Grytczuk--Przybyło and disprove others.
Comments13 pages, no figures
