Direct sum subset decompositions of<i>Z</i>

Swenson, Carl

doi:10.2140/pjm.1974.53.629

Cited by 12 publications

(4 citation statements)

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“…In conclusion we remark that Sands' problem for the special case of cyclic groups G was raised earlier by C. Swenson [8]. Our methods do not yield counterexamples for this case; thus Swenson's problem remains open.…”

Section: H-\°mentioning

confidence: 59%

Solution to a problem of A. D. Sands

Fraser

Gordon

1979

Glasgow Math. J.

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Section: H-\°mentioning

confidence: 59%

Solution to a problem of A. D. Sands

Fraser

Gordon

1979

Glasgow Math. J.

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“…While no complete characterization of the tiles of Z is available, it is well known that being a tile imposes strong conditions of arithmetic flavor on the structure of a set (e.g. see [5,8,4,9,10]). The Coven-Meyerowitz Conjecture states that being a tile is equivalent to a purely arithmetic condition on the set (see below, in Section 2).…”

Section: Introductionmentioning

confidence: 99%

Arithmetic sets in groups

Akhmedov

Fulghesu

2018

Math. Z.

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“…Tilings of this type have been extensively studied (cf. [2][3][4]6,7,9,10,12,14,15,[17][18][19]21,22]) but one still seems to know rather little about them. For example, a basic conjecture of Coven and Meyerowitz remains unaddressed [3,4], as well as the "quasiperiodic conjecture" of Hajós [5,13,20,18].…”

Section: Introductionmentioning

confidence: 99%

Tilings of the integers can have superpolynomial periods

Steinberger

2009

Combinatorica

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Let A be a finite set of integers. We say that A tiles the integers if there is a set T ⊆ Z such that {t + A : t ∈ T } forms a disjoint partition of the integers. It has long been known that such a set T must be periodic. The question is to determine how long the period of T can become as a function of the diameter of the set A. The previous best lower bound, due to Kolountzakis [7], shows that the period of T can grow as fast as the square of the diameter of A. In this paper we improve Kolountzakis' lower bound by showing that the period of T can in fact grow faster than any power of the diameter of A.

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Direct sum subset decompositions ofZ

Cited by 12 publications

References 4 publications

Solution to a problem of A. D. Sands

Solution to a problem of A. D. Sands

Arithmetic sets in groups

Tilings of the integers can have superpolynomial periods

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