Decomposition of the integers as a direct sum of two subsets

“…This in turn leads to a structure theorem for bounded tiles. It was also observed in [15] that the "tiling implies spectrum" part of Fuglede's conjecture for compact sets in R would follow from a conjecture of Tijdeman [20] concerning factorization of finite cyclic groups; however, Tijdeman's conjecture is now known to fail without additional assumptions (see [13] for a discussion). See also [16], [1] for partial results on the related problem of characterizing all tilings of Z by a finite set, and [15], [18] for a classification of domains in R n which have L + Z n as a spectrum for some finite set L. Another recent result [19] is that sets which tile [0, ∞) by translations are spectral sets.…”

Section: An Orthonormal Basis For L 2 (ω) (11)mentioning

confidence: 99%

Fuglede’s conjecture for a union of two intervals

Łaba

2001

Proc. Amer. Math. Soc.

107

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Abstract. We prove that a union of two intervals in R is a spectral set if and only if it tiles R by translations. The results A Borel set Ω ⊂ Rn of positive measure is said to tile R n by translations if there is a discrete set T ⊂ R n such that, up to sets of measure 0, the sets Ω + t, t ∈ T, are disjoint and t∈T (Ω + t) = R n . We may rescale Ω so that |Ω| = 1. We say that Λ = {λ k : k ∈ Z} ⊂ R n is a spectrum for Ω ifA spectral set is a domain Ω ⊂ R n such that (1.1) holds for some Λ. Fuglede [2] conjectured that a domain Ω ⊂ R n is a spectral set if and only if it tiles R n by translations, and proved this conjecture under the assumption that either Λ or T is a lattice. The conjecture is related to the question of the existence of commuting self-adjoint extensions of the operators −i Recently there has been significant progress on the special case of the conjecture when Ω is assumed to be convex [10], [3], [4], and in particular the 2-dimensional convex case appears to be nearly resolved [5]. The non-convex case is considerably more complicated, and is not understood even in dimension 1. The strongest results yet in that direction seem to be those of Lagarias and Wang [14], [15], who proved that all tilings of R by a bounded region must be periodic, and that the corresponding translation sets are rational up to affine transformations. This in turn leads to a structure theorem for bounded tiles. It was also observed in [15] that the "tiling implies spectrum" part of Fuglede's conjecture for compact sets in R would follow from a conjecture of Tijdeman [20] concerning factorization of finite cyclic groups; however, Tijdeman's conjecture is now known to fail without additional assumptions (see [13]

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“…As an immediate corollary, one obtains the decidability of the tiling of the discrete line by a single pattern. Despite that, its algorithmic complexity remains open although a lot of efforts have been dedicated to study bases for the integers [6,21]. Moreover, the periodicity of the positions set B raises the question of the characterization of sets A and B such that A ⊕ B = Z/nZ.…”

Section: Introductionmentioning

confidence: 99%

Tiling an Interval of the Discrete Line

Bodini

Rivals

2006

Combinatorial Pattern Matching

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Abstract. We consider the problem of tiling a segment {0, . . . , n} of the discrete line. More precisely, we ought to characterize the structure of the patterns that tile a segment and their number. A pattern is a subset of N. A tiling pattern or tile for {0, . . . , n} is a subset A ∈ P(N) such that there exists B ∈ P(N) and such that the direct sum of A and B equals {0, . . . , n}. This problem is related to the difficult question of the decomposition in direct sums of the torus Z/nZ (proposed by Minkowski). Using combinatorial and algebraic techniques, we give a new elementary proof of Krasner factorizations. We combinatorially prove that the tiles are direct sums of some arithmetic sequences of specific lengths. Besides, we show there are as many tiles whose smallest tilable segment is {0, . . . , n} as tiles whose smallest tilable segment is {0, . . . , d}, for all strict divisors d of n. This enables us to exhibit an optimal linear time algorithm to compute for a given pattern the smallest segment that it tiles if any, as well as a recurrence formula for counting the tiles of a segment.

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Decomposition of the integers as a direct sum of two subsets

Cited by 46 publications

References 0 publications

The Favard length of product Cantor sets

The Favard length of product Cantor sets

Fuglede’s conjecture for a union of two intervals

Tiling an Interval of the Discrete Line

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