TY - JOUR

T1 - Optimal Arithmetic Structure in Exponential Riesz Sequences

AU - Londner, Itay

N1 - Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2020/2/1

Y1 - 2020/2/1

N2 - We consider exponential systems E(Λ)={eiλt}λ∈Λ for Λ ⊂ Z. It has been previously shown by Londner and Olevskii (Stud Math 255(2):183–191, 2014) that there exists a subset of the circle, of positive Lebesgue measure, so that every set Λ which contains, for arbitrarily large N, an arithmetic progressions of length N and step ℓ= O(Nα) , α< 1 , cannot be a Riesz sequence in the L2 space over that set. On the other hand, every set admits a Riesz sequence containing arbitrarily long arithmetic progressions of length N and stepℓ= O(N). In this paper we show that every set S⊂ T of positive measure belongs to a unique class, defined through the optimal growth rate of the step of arithmetic progressions with respect to the length that can be found in Riesz sequences in the space L2(S). We also give a partial geometric description of each class.

AB - We consider exponential systems E(Λ)={eiλt}λ∈Λ for Λ ⊂ Z. It has been previously shown by Londner and Olevskii (Stud Math 255(2):183–191, 2014) that there exists a subset of the circle, of positive Lebesgue measure, so that every set Λ which contains, for arbitrarily large N, an arithmetic progressions of length N and step ℓ= O(Nα) , α< 1 , cannot be a Riesz sequence in the L2 space over that set. On the other hand, every set admits a Riesz sequence containing arbitrarily long arithmetic progressions of length N and stepℓ= O(N). In this paper we show that every set S⊂ T of positive measure belongs to a unique class, defined through the optimal growth rate of the step of arithmetic progressions with respect to the length that can be found in Riesz sequences in the space L2(S). We also give a partial geometric description of each class.

KW - Arithmetic progressions

KW - Interpolation sets

KW - Riesz sequences

UR - http://www.scopus.com/inward/record.url?scp=85077519128&partnerID=8YFLogxK

U2 - 10.1007/s00041-019-09706-9

DO - 10.1007/s00041-019-09706-9

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AN - SCOPUS:85077519128

SN - 1069-5869

VL - 26

JO - Journal of Fourier Analysis and Applications

JF - Journal of Fourier Analysis and Applications

IS - 1

M1 - 1

ER -