In this work, we study the non-spectrality of the self-affine measure μM,D{\mu_{M,D}} generated by an expanding integer matrix M∈M2⁢(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,(α1,α2)t,(β1,β2)t,(-α1-β1,-α2-β2)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 α1⁢β2-α2⁢β1≠0{\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\neq 0}. Let η=max⁡{s:2s|(α1⁢β2-α2⁢β1)}{\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 L2⁢(μM,D){L^{2}(\mu_{M,D})} contains at most 22⁢(η+1){2^{2(\eta+1)}} mutually orthogonal exponential functions, and the number 22⁢(η+1){2^{2(\eta+1)}} is the best. However, the number is strictly less than 22⁢(η+1){2^{2(\eta+1)}} if η≥3{\eta\geq 3}, and it is related to the order of the matrix M .

中文翻译：

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μM的基数，平面四位数的D-正交指数

在这项工作中，我们研究了由扩展整数矩阵 M∈M2⁢(ℤ){M\in M_{2}( \mathbb{Z})} 与 det⁡(M)∉2⁢ℤ{\det(M)\notin 2\mathbb{Z}} 和整数数字集 D={(0,0)t,(α1, α2)t,(β1,β2)t,(-α1-β1,-α2-β2)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{\}} 与 α1⁢β2-α2⁢β1≠0{\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\neq 0}。令 η=max⁡{s:2s|(α1⁢β2-α2⁢β1)}{\eta=\max\{s:2^{s}|(\alpha_{1}\beta_{2}-\alpha_ {2}\beta_{1})\}}。我们证明如果 0≤η≤2{0\leq\eta\leq 2}，那么 L2⁢(μM,D){L^{2}(\mu_{M,D})} 最多包含 22⁢( η+1){2^{2(\eta+1)}} 相互正交的指数函数，数 22⁢(η+1){2^{2(\eta+1)}} 是最好的。然而，