On Steinhaus sets

Srivastava, S. M.; Thangadurai, R.

doi:10.1016/j.exmath.2005.01.011

Cited by 4 publications

(4 citation statements)

References 8 publications

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“…Kolountzakis [10,11] and Kolountzakis and Wolff [15] proved that any measurable set in the plane that has the measurable Steinhaus property must necessarily have very slow decay at infinity (it is easy to see that any such set must have measure 1). In [15], it was also shown that there can be no measurable Steinhaus sets in dimension d 3 (with the obvious definition) a fact that was also shown later by Kolountzakis and Papadimitrakis [14] by a very different method (see also [3,4,17,21]). Kolountzakis [12] looks at the case where the linear transformation ρ is only required to take on finitely many values.…”

Section: Introductionmentioning

confidence: 74%

“…In [15] it was also shown that there can be no measurable Steinhaus sets in dimension d ≥ 3 (with the obvious definition) a fact that was also shown later by Kolountzakis and Papadimitrakis [14] by a very different method. See also [3,18,4,22]. Kolountzakis [12] looks at the case where the linear transformation ρ is only required to take on finitely many values.…”

Section: Introductionmentioning

confidence: 99%

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Measurable Steinhaus sets do not exist for finite sets or the integers in the plane

Kolountzakis

Papadimitrakis

2017

Bull. London Math. Soc.

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Abstract. A Steinhaus set S ⊆ Rd for a set A ⊆ R d is a set such that S has exactly one point in common with τA, for every rigid motion τ of R d . We show here that if A is a finite set of at least two points then there is no such set S which is Lebesgue measurable.An old result of Komjáth says that there exists a Steinhaus set for A = Z × {0} in R 2 . We also show here that such a set cannot be Lebesgue measurable.

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Section: Introductionmentioning

confidence: 74%

Section: Introductionmentioning

confidence: 99%

Measurable Steinhaus sets do not exist for finite sets or the integers in the plane

Kolountzakis

Papadimitrakis

2017

Bull. London Math. Soc.

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“…In [22] it was also shown that there can be no measurable Steinhaus sets in dimension d ≥ 3 (tiling with all rotates ρZ d , where ρ is in the full orthogonal group) a fact that was also shown later by Kolountzakis and Papadimitrakis [20] by a very different method. See also [5,25,6,29]. Kolountzakis [18] looks at the case where we are only asking for our set to tile with finitely many lattices, not all rotates as in the original problem, which we are also doing in this paper.…”

Section: Smentioning

confidence: 99%

Functions tiling with several lattices

Kolountzakis¹,

Papageorgiou²

2021

Preprint

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A. We study the problem of finding a function f with "small support" that simultaneously tiles with finitely many lattices Λ 1 , . . . , Λ N in d-dimensional Euclidean spaces. We prove several results, both upper bounds (constructions) and lower bounds on how large this support can and must be. We also study the problem in the setting of finite abelian groups, which turns out to be the most concrete setting. Several open questions are posed.

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“…In [19] it was also shown that there can be no measurable Steinhaus sets in dimension d ≥ 3 (tiling with all rotates ρZ d , where ρ is in the full orthogonal group) a fact that was also shown later by Kolountzakis and Papadimitrakis [17] by a very different method. See also [3,24,4,27]. Kolountzakis [15] looks at the case where we are only asking for our set to tile with finitely many lattices, not all rotates as in the original problem, which we are also doing in this paper.…”

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confidence: 99%

Simultaneous tiling

Kolountzakis¹

2022

Preprint

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MIHAIL N. KOLOUNTZAKIS A. We discuss problems of simultaneous tiling. This means that we have an object (set, function) which tiles space with two or more different sets of translations. The most famous problem of this type is the Steinhaus problem which asks for a set simultaneously tiling the plane with all rotates of the integer lattice as translation sets. Dedicated to the memory of Dimitris GatzourasC 1. Tiling by translation. 1 2. Tiling in Fourier space. 2 3. The Steinhaus tiling problem. 3 4. The Steinhaus problem in Fourier space. 4 5. Allowing functions instead of sets in the Steinhaus problem. 4 6. Small diameter of the support: lower bounds. 5 7. A case of large diameter. 6 8. Small volume of the support. 7 9. Small length of the support in d = 1. 9 10. Very small diameter of the support with relations among the lattices. 9 11. Almost matching upper and lower bounds for the diameter, d = 1. 11 References 12 §1. Tiling by translation. For the purposes of this paper 1 tiling is by translation only [16]. We have an object T (the tile) which may be a set or a function on some abelian group G (usually the Euclidean space but it may be Z d or a finite group) which we are F . An L-shaped tile. The red point is the origin translating around by a set of translations Λ, in such a way that everything in the group

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On Steinhaus sets

Cited by 4 publications

References 8 publications

Measurable Steinhaus sets do not exist for finite sets or the integers in the plane

Measurable Steinhaus sets do not exist for finite sets or the integers in the plane

Functions tiling with several lattices

Simultaneous tiling

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