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arXiv · 2607.06113

Solution to a conjecture of Alon, D\k{e}bski, Grytczuk and Przyby\l{}o on fixed-cardinality arithmetic progressions

Yaping Mao·Zhao Wang·Meiqin Wei·Gang Yang
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Abstract

Fix a positive integer $n$, and put $B_d=\{d,2d,\ldots,nd\}$. Let $M_k(n)$ be the least integer $m$ for which one translate of each of $B_1,\ldots,B_k$ can be placed pairwise disjointly in $[m]$. We prove that, for every $\eps\in(0,1)$ and all sufficiently large $k$, one has $M_k(n)\le n\lceil(1+\eps)k\rceil$. Since the trivial counting bound gives $M_k(n)\ge nk$, it follows that $M_k(n)=(1+o(1))nk$ for every fixed $n$. This confirms a conjecture of Alon, D\k{e}bski, Grytczuk and Przyby\l{}o on prescribed-difference packings of fixed-cardinality arithmetic progressions.

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Yaping Mao, Zhao Wang, Meiqin Wei, Gang Yang. 2026-07-07. Solution to a conjecture of Alon, D\k{e}bski, Grytczuk and Przyby\l{}o on fixed-cardinality arithmetic progressions. https://arxiv.org/abs/2607.06113

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