DOIAbstract
AbstractA striking conjecture of Fraenkel asserts that every decomposition of Z>0 into m⩾3 sets {⌊αin+βi⌋}n∈Z>0 with αi and βi real, αi>1 and αi's distinct for i=1,…,m satisfies{α1,…,αm}=2m−12k:0⩽k<m.Fraenkel's conjecture was proved by Morikawa if m=3 and, under some condition, if m=4. Proofs in terms of balanced sequences have been given for m=3 by the author and for m=4 by Altman, Gaujal and Hordijk. In the present paper we use the latter approach to establish Fraenkel's conjecture for m=5 and for m=6
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