Covering three-tori with cubes

2021/12/22 by Bogdanov, Ilya, Grigoryan, Oleg, Zhukovskii, Maksim
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.12202

Abstract

Let μ(ε) be the minimum number of cubes of side ε needed to cover the unit three-torus [ℝ/ℤ]³. We prove new lower and upper bounds for μ(ε) and find the exact value for all ε≥(7)/(15) and all ε∈[(1)/(r+1/(r²+r+1)),(1)/(r-1/(r²-1))) for any integer r≥ 2.

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