Covering the ternary cube by binary subcubes · arXivDesk2608.13252Aug 13, 2026
Covering the ternary cube by binary subcubes
Peiru Kuang, Yan Wang
Abstract
For an integer n≥0, let f(n) be the minimum number of subcubes of Z3n of the form A1×⋯×An
, where
for every
, whose union covers
. A simple counting argument gives
f(n)≥(3/2)n , while
f(n)=O(n(3/2)n) by random construction. We prove that
f(n)≤2(3/2)n−1 , answering a problem of Imre Leader. We also show that
f(n)/(3/2)n is nondecreasing and there exists a constant
such that
f(n)=(C3+o(1))(3/2)n where
1.62227<C3≤2 .