A counterexample to Dutkay–Jorgensen conjecture.

For an expanding integer matrix M ∈ M n ( Z ) and two finite subsets D , S ⊂ R n of the same cardinality, the concept of compatible pair ( M − 1 D , S ) (or Hadamard triple ( M , D , S ) ) plays an important role in the spectrality of self-affine measure μ M , D . It is known that ( M − 1 D , S ) is a compatible pair if and only if ( M ⁎ − 1 S , D ) is a compatible pair. An old duality conjecture of Dutkay and Jorgensen states that under the condition of compatible pair ( M − 1 D , S ) , μ M , D is a spectral measure if and only if μ M ⁎ , S is. In this paper, we construct an example of compatible pair ( M − 1 D , S ) to illustrate that the self-affine measure μ M , D is a spectral measure but the self-affine measure μ M ⁎ , 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. [ABSTRACT FROM AUTHOR]