Covering the positive integers by disjoint sets of the form {[nα + β]: n = 1, 2,…}
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This article is published in Journal of Combinatorial Theory, Series A.The article was published on 1973-11-01 and is currently open access. It has received 74 citations till now. The article focuses on the topics: Disjoint sets.read more
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Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences
TL;DR: A two-dimensional generalization of Sturmian sequences corresponding to an approximation of a plane is studied, which deduces a new way of computing the rectangle complexity function and provides an upper bound on the number of frequencies of rectangular factors of given size.
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Beatty Sequences, Continued Fractions, and Certain Shift Operators
TL;DR: In this article, the characteristic function of the infinite sequence [n θ] itself was generated from the finite sequence [ n θ]-[nθ] by using shift operators.
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Balanced sequences and optimal routing
TL;DR: In this article, it was shown that the optimal deterministic routing in stochastic event graphs is such a sequence, where each letter is distributed as "evenly" as possible and appears with a given rate.
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Balances for fixed points of primitive substitutions
TL;DR: This paper generalizes the notion of a measure of balance for an infinite sequence and shows that the asymptotic behaviour of this measure is in part ruled by the spectrum of the incidence matrix associated with the substitution.
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Fraenkel's conjecture for six sequences
TL;DR: The present paper uses the latter approach to establish Fraenkel's conjecture for m=5 and for m-6, and proves in terms of balanced sequences that every decomposition of Z >0 into m⩾3 sets satisfies αi and βi real, αi>1 and αi's distinct for i=1,…,m.
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The Bracket Function and Complementary Sets of Integers
TL;DR: In this article, it was shown that if α and β are positive irrational numbers satisfying 1 then the sets [nα], [nβ], n = 1, 2, …, are complementary with respect to the set of all positive integers, if and only if β and α are irrational.
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Complementing and exactly covering sequences
TL;DR: A method of reduction is developed which, given a complementing system of m sequences, leads under certain conditions to a derived complementingSystem of m − 1 sequences.