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

Kolountzakis, Mihail N.; Papadimitrakis, Michael

doi:10.1112/blms.12069

Cited by 5 publications

(3 citation statements)

References 18 publications

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“…It is known that every set of cardinality 3, 4, 5 or 7 is a Jackson set (see [3]). It is also known that for every finite set K ⊂ R k having at least two elements there is no measurable sets that intersect each congruent copy of K in exactly one point [9]. Now, we show that if a set of cardinality at least two has the Pompeiu property, then it is a Jackson set.…”

Section: Introductionmentioning

confidence: 73%

The discrete Pompeiu problem on the plane

2017

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We say that a finite subset E of the Euclidean plane R 2 has the discrete Pompeiu property with respect to isometries (similarities), if, whenever f : R 2 → C is such that the sum of the values of f on any congruent (similar) copy of E is zero, then f is identically zero. We show that every parallelogram and every quadrangle with rational coordinates has the discrete Pompeiu property w.r.t. isometries. We also present a family of quadrangles depending on a continuous parameter having the same property. We investigate the weighted version of the discrete Pompeiu property as well, and show that every finite linear set with commensurable distances has the weighted discrete Pompeiu property w.r.t. isometries, and every finite set has the weighted discrete Pompeiu property w.r.t. similarities.

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Section: Introductionmentioning

confidence: 73%

The discrete Pompeiu problem on the plane

2017

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“…Kolountzakis [17,16] and Kolountzakis and Wolff [22] proved that any measurable set in the plane that has the measurable Steinhaus property must necessarily have very slow decay at infinity (any such set must have measure 1). 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].…”

Section: Smentioning

confidence: 79%

Functions tiling with several lattices

Kolountzakis¹,

Papageorgiou²

2021

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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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“…Kolountzakis [14,13] and Kolountzakis and Wolff [19] proved that any measurable set in the plane that has the measurable Steinhaus property must necessarily have very slow decay at infinity (any such set must have measure 1). 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].…”

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confidence: 79%

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

Cited by 5 publications

References 18 publications

The discrete Pompeiu problem on the plane

The discrete Pompeiu problem on the plane

Functions tiling with several lattices

Simultaneous tiling

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