Riesz bases of exponentials on unbounded multi-tiles

Abstract

We prove the existence of Riesz bases of exponentials of L 2 (Ω), provided that Ω ⊂ R d is a measurable set of finite and positive measure, not necessarily bounded, that satisfies a multi-tiling condition and an arithmetic property that we call admissibility. This property is satisfied for any bounded domain, so our results extend the known case of bounded multi-tiles. We also extend known results for submulti-tiles and frames of exponentials to the unbounded case.

On construction of bounded sets not admitting a general type of Riesz spectrum

“…They also showed that the boundedness of S is essential by constructing an unbounded 2-tile set S ⊂ R with respect to Z which does not admit a (2, Z)-structured Riesz spectrum. Nevertheless, for unbounded multi-tiles S ⊂ R d with respect to a lattice Γ, Cabrelli and Carbajal [4] were able to provide a sufficient condition for S to admit a structured Riesz spectrum. Recently, Cabrelli et al [5] found a necessary and sufficient condition for a multi-tile S ⊂ R d of finite positive measure to admit a structured Riesz spectrum, which is given in terms of the Bohr compactification of the tiling lattice Γ.…”

Riesz bases of exponentials and the Bohr topology

“…In [2], there was an example of a k-tile with respect to Z in R, that was not admissible but had a structured Riesz basis. Here, using Proposition 4.6, we construct a different example using ε-Kronecker sets.…”

“…An example found in [1] shows that an unbounded multi-tile need not support structured Riesz bases. The question of the existence of unbounded multi-tiles with structured bases of exponentials was recently addressed in [2] where it was proven that admissible multi-tiles always have a structured Riesz basis. The class of admissible multi-tiles (see Example 4.3 for the definition) contains all the bounded multi-tiles, as well as a large class of unbounded multi-tiles.…”

“…The class of admissible multi-tiles (see Example 4.3 for the definition) contains all the bounded multi-tiles, as well as a large class of unbounded multi-tiles. However, admissibility is only a sufficient condition, as seen in [2], and thus it is an open question as to exactly which multi-tiles support a structured Riesz basis of exponentials.…”

“…Take P = M ([0, 1) d ) as the fundamental domain. As in[2], we say Ω ⊆ R d of finite measure is admissible for Λ if there is some v ∈ R d and integer n such that the numbers {v • λ : λ ∈ Λ ω } are distinct integers modulo n for a.e. ω. Redefining Q by omitting a set of measure zero, this ensures that if Λ ω = {λ i (ω) : i = 1, ..., k}, then v • (λ i − λ j ) ∈ Z nZ for any i = j and all ω ∈ Q.Hence1 − e v/n (λ j − λ ℓ ) ≥ |1 − exp i2π/n| := ε > 0 for all j = ℓ.…”