Non-spectral problem for the planar self-affine measures

By Jingcheng Liu, Xin-Han Dong, Jian-Lin Li. 2016

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Abstract In this paper, we consider the non-spectral problem for the planar self-affine measures μ M , D generated by an expanding integer matrix M ∈ M 2 ( Z ) and a finite digit set D ⊂ Z 2 . Let p ≥ 2 be a positive integer, E p 2 : = 1 p { ( i , j ) t : 0 ≤ i , j ≤ p − 1 } and Z D 2 : = { x ∈ [ 0 , 1 ) 2 : ∑ d ∈ D e 2 π i 〈 d , x 〉 = 0 } . We show that if ∅ ≠ Z D 2 ⊂ E p 2 ∖ { 0 } and gcd ⁡ ( det ⁡ ( M ) , p ) = 1 , then there exist at most p 2 mutually orthogonal exponential functions in L 2 ( μ M , D ) . In particular, if p is a prime, then the number p 2 is the best.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f07b9338c952f32777b829e7435b7a71f50c09ba