A counterexample to Dutkay–Jorgensen conjecture

For an expanding integer matrix M∈Mn(Z)M∈Mn(Z) and two finite subsets D,S⊂RnD,S⊂Rn of the same cardinality, the concept of compatible pair (M−1D,S)(M−1D,S) (or Hadamard triple (M,D,S)(M,D,S)) plays an important role in the spectrality of self-affine measure μM,DμM,D. It is known that (M−1D,S)(M−1D,S) is a compatible pair if and only if (M⁎−1S,D)(M⁎−1S,D) is a compatible pair. An old duality conjecture of Dutkay and Jorgensen states that under the condition of compatible pair (M−1D,S)(M−1D,S), μM,DμM,D is a spectral measure if and only if μM⁎,SμM⁎,S is. In this paper, we construct an example of compatible pair (M−1D,S)(M−1D,S) to illustrate that the self-affine measure μM,DμM,D is a spectral measure but the self-affine measure μM⁎,SμM⁎,S is not. This disproves the above-mentioned conjecture of Dutkay and Jorgensen, and clarifies certain dual relation on the spectrality of self-affine measures.