Nonspectrality of Certain Self-Affine Measures on ℝ 3

Abstract EN

We will determine the nonspectrality of self-affine measure μ B , D corresponding to B = diag [ p 1 , p 2 , p 3 ] ( p 1 ∈ ( 2 ℤ + 1 ) ∖ { ± 1 } , p 2 ∈ 2 ℤ ∖ { 0 } ) , and D = { 0 , e 1 , e 2 , e 3 } in the space ℝ 3 is supported on T ( B , D ) , where e 1 , e 2 , and e 3 are the standard basis of unit column vectors in ℝ 3 , and there exist at most 4 mutually orthogonal exponential functions in L 2 ( μ B , D ) , where the number 4 is the best.

This generalizes the known results on the spectrality of self-affine measures.