Fuglede’s Conjecture for a Union of Two Intervals

  title={Fuglede’s Conjecture for a Union of Two Intervals},
  author={Izabella Laba},
We prove that a union of two intervals in R is a spectral set if and only if it tiles R by translations. 1. The results A Borel set Ω ⊂ R of positive measure is said to tile R by translations if there is a discrete set T ⊂ R such that, up to sets of measure 0, the sets Ω + t, t ∈ T, are disjoint and ⋃ t∈T (Ω + t) = R . We may rescale Ω so that |Ω| = 1. We say that Λ = {λk : k ∈ Z} ⊂ R is a spectrum for Ω if {ek}k∈Z is an orthonormal basis for L(Ω). (1.1) A spectral set is a domain Ω ⊂ R such… CONTINUE READING
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