缺项斜积的正则性、定量偏差与非刚性
Regularity, quantitative deviation, and non-rigidity of a lacunary skew product
AI总结：
该研究针对无理数α及其连分数收敛项分母构造的函数h与斜积f，分析其正则性、偏差性质与非刚性，明确了f的连续性、Lipschitz性及Diophantine条件下的偏差特征。
AI中文摘要：
设α为无理数，q_j为其连分数收敛项的分母。研究函数h(x)=∑_{j≥1}cos(2πq_jx)/q_j及斜积f(x,y)=(x+α,y+h(x)) mod ℤ²。函数h是所有低于1指数的Hölder连续函数，傅里叶论证表明其非Lipschitz。映射f是带旋转向量(α,0)的环面伪旋转，但既无有界平均运动也无C⁰刚性。若α满足Diophantine条件DC(τ)，则当τ>1时f有(C,1−1/τ)偏差；当τ=1时，对任意0<δ<1有(C_δ,δ)偏差，但δ=0时无此偏差。
英文摘要：
Let $α$ be irrational and let $q_j$ be the denominators of its continued-fraction convergents. We study the function \[ h(x)=\sum_{j\geq1}\frac{\cos(2πq_jx)}{q_j} \] and the skew product \[f(x,y)=(x+α,y+h(x))\quad\mathrm{mod}\quad\mathbb{Z}^2.\] The function $h$ is Hölder continuous of every exponent below one. A Fourier argument shows that $h$ is not Lipschitz. The map $f$ is a toral pseudo-rotation with rotation vector $(α,0)$, but it has neither bounded mean motion nor $C^0$-rigidity. Suppose $α$ satisfies the Diophantine condition $\mathrm{DC}(τ)$. Then, $f$ has $(C,1-1/τ)$-deviation when $τ>1$; and it has $(C_δ,δ)$-deviation for every $0<δ<1$, but not for $δ=0$ when $τ=1$.
