For $$\tau >3$$ τ > 3 and $$\alpha \in \mathbb {R}$$ α ∈ R , let T be a skew product map of the form $$T(x_1,x_2)=(x_1+\alpha ,x_2+h(x_1))$$ T ( x 1 , x 2 ) = ( x 1 + α , x 2 + h ( x 1 ) ) on $$\mathbb {T}^2$$ T 2 over a rotation of the circle, for which h is of zero topological degree and of class $$C^{\tau }$$ C τ . We prove that for a measure-theoretically generic set of $$\alpha $$ α , such a $$C^{\tau }$$ C τ skew product map T is strongly orthogonal to the Möbius function. Moreover, we establish the strong orthogonality between the Möbius function and some irregular $$C^{\tau }$$ C τ skew product map on $$\mathbb {T}^2$$ T 2 . The key tools used are Fourier analysis and exponential sums estimate concerning the Möbius function.

中文翻译：

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莫比乌斯函数和偏斜积之间的强正交性

对于 $$\tau >3$$ τ > 3 和 $$\alpha \in \mathbb {R}$$ α ∈ R ，令 T 为 $$T(x_1,x_2)=( x_1+\alpha ,x_2+h(x_1))$$ T ( x 1 , x 2 ) = ( x 1 + α , x 2 + h ( x 1 ) ) 在 $$\mathbb {T}^2$$ T 2 在圆的旋转上，其中 h 的拓扑度为零并且属于 $$C^{\tau }$$ C τ 类。我们证明，对于 $$\alpha $$ α 的度量理论上通用的集合，这样的 $$C^{\tau }$$ C τ skew product map T 与 Möbius 函数强正交。此外，我们建立了莫比乌斯函数与 $$\mathbb {T}^2$$ T 2 上的一些不规则 $$C^{\tau }$$ C τ skew 产品图之间的强正交性。使用的关键工具是关于莫比乌斯函数的傅立叶分析和指数和估计。