For every fixed n, the minimum interval length needed to pack k disjoint translates of n-term arithmetic progressions with differences 1 through k equals (1+o(1))nk.
Johnson, Perfect rhythmic tilings, lecture at the MaMuX seminar, IRCAM, 2004
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Solution to a conjecture of Alon, D\k{e}bski, Grytczuk and Przyby\l{}o on fixed-cardinality arithmetic progressions
For every fixed n, the minimum interval length needed to pack k disjoint translates of n-term arithmetic progressions with differences 1 through k equals (1+o(1))nk.