The cardinality of μ M,D ‐orthogonal exponentials for the planar four digits

Abstract

Abstract In this work, we study the non-spectrality of the self-affine measure μ M , D {\mu_{M,D}} generated by an expanding integer matrix M ∈ M 2 ⁢ ( ℤ ) {M\in M_{2}(\mathbb{Z})} with det ⁡ ( M ) ∉ 2 ⁢ ℤ {\det(M)\notin 2\mathbb{Z}} and the integer digit set D = { ( 0 , 0 ) t , ( α 1 , α 2 ) t , ( β 1 , β 2 ) t , ( - α 1 - β 1 , - α 2 - β 2 ) t } D=\bigl{\{}(0,0)^{t},(\alpha_{1},\alpha_{2})^{t},(\beta_{1},\beta_{2})^{t},(-% \alpha_{1}-\beta_{1},-\alpha_{2}-\beta_{2})^{t}\bigr{\}} with α 1 ⁢ β 2 - α 2 ⁢ β 1 ≠ 0 {\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\neq 0} . Let η = max ⁡ { s : 2 s | ( α 1 ⁢ β 2 - α 2 ⁢ β 1 ) } {\eta=\max\{s:2^{s}|(\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1})\}} . We show that if 0 ≤ η ≤ 2 {0\leq\eta\leq 2} , then L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} contains at most 2 2 ⁢ ( η + 1 ) {2^{2(\eta+1)}} mutually orthogonal exponential functions, and the number 2 2 ⁢ ( η + 1 ) {2^{2(\eta+1)}} is the best. However, the number is strictly less than 2 2 ⁢ ( η + 1 ) {2^{2(\eta+1)}} if η ≥ 3 {\eta\geq 3} , and it is related to the order of the matrix M.