Interpolating classical partitions of the set of positive integers
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Interpolating classical partitions of the set of positive integers Jared Krandel1 · Weiru Chen1 Received: 4 November 2018 / Accepted: 17 July 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We construct an easily described family of partitions of the positive integers into n disjoint sets with essentially the same structure for every n ≥ 2. In a special case, it interpolates between the Beatty φ1 + φ12 = 1 partitioning (n = 2) and the 2-adic partitioning in the limit as n → ∞. We then analyze how membership of elements in the sets of one partition relates to membership in the sets of another. We investigate in detail the interactions of two Beatty partitions with one another and the interactions of the φ Beatty partition mentioned above with its “extension” to three sets given by the construction detailed in the first part. In the first case, we obtain detailed results, whereas in the second case we place some restrictions on the interaction but cannot obtain exhaustive results. Keywords Partitions · Beatty · Golden ratio · Fractional parts · Sequences Mathematics Subject Classification 05A18 · 11B05
1 Introduction Beatty’s x1 + 1y = 1 theorem [1] provides uncountably many partitionings of the set of positive integers. Among these are partitionings that encode winning strategies for variations of Wythoff’s game [16]. Particularly relevant is the partition given by 1 1 + 2 =1 φ φ
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Weiru Chen [email protected] Jared Krandel [email protected]
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Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, USA
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J. Krandel, W. Chen
with the members of the two distinct sequences a(k) = kφ and b(k) = kφ 2 giving the two disjoint sets of the partition. Another significant partition, this time into infinitely many sets, is the 2-adic partition. Here, the sets are 2e · M for e = 0, 1, 2, . . ., where M is the set of all positive odd integers. This partitioning is, for example, the key to showing that the partial sums of the harmonic series are (with the trivial exception) never integral. See [p. 32, problem 36] of [10]. We construct an easily described family of partitions of the positive integers into n disjoint sets with essentially the same structure for every n ≥ 2. The construction goes as follows. Fix n ≥ 2 and choose a non-decreasing, integer-valued sequence l such that the difference of any two consecutive values of l lie in the set n 2n−i . G = 2n−1 , 2n−1 + 2n−2 , . . . , i=1
Define D1 to be the set of integers in the range of l and call D2 the set of integers 2n−2 “away” from any integer in D1 . That is, D2 = {k − 2n−2 : k ∈ D1 } ∪ {k + 2n−2 : k ∈ D1 }. In a similar fashion, define D3 to be the integers 2n−3 “away” from those in D2 , and continue defining sets D j in this fashion up to Dn , which is the set of integers 1 “away” from Dn−1 . The sets D1 , D2 , . . . , Dn form the partition. The choice of sequence l defining D1 determines the structure of the partition. In a special case, it interpolates between the Beatty golde
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