2021/08/17 by Dae Gwan Lee, Lee, Dae Gwan
Mathematics · #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.2108.07760
Despite the recent advances in the theory of exponential Riesz bases, it is yet unknown whether there exists a set S ⊂ ℝd which does not admit a Riesz spectrum, meaning that for every Λ⊂ ℝd the set of exponentials e2πi λ⋅ x with λ∈Λ is not a Riesz basis for L2(S). As a meaningful step towards finding such a set, we construct a set S ⊂ [-(1)/(2), (1)/(2)] which does not admit a Riesz spectrum containing a nonempty periodic set with period belonging in αℚ+ for any fixed constant α> 0, where ℚ+ denotes the set of all positive rational numbers. In fact, we prove a slightly more general statement that the set S does not admit a Riesz spectrum containing arbitrarily long arithmetic progressions with a fixed common difference belonging in αℕ. Moreover, we show that given any countable family of separated sets Λ1, Λ2, … ⊂ ℝ with positive upper Beurling density, one can construct a set S ⊂ [-(1)/(2), (1)/(2)] which does not admit the sets Λ1, Λ2, … as Riesz spectrum. An interesting consequence of our results is the following statement. There is a set V ⊂ [-(1)/(2), (1)/(2)] with arbitrarily small Lebesgue measure such that for any N ∈ ℕ and any proper subset I of \ 0, …, N-1 \, the set of exponentials e2πi k x with k ∈ ∪n ∈ I (Nℤ + n) is not a frame for L2(V). The results are based on the proof technique of Olevskii and Ulanovskii in 2008.