Solution to a conjecture of Alon, Dębski, Grytczuk and Przybyło on fixed-cardinality arithmetic progressions

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, Dębski, Grytczuk and Przybyło on prescribed-difference packings of fixed-cardinality arithmetic progressions.