AI 中文总结
本文利用有限分辨率下的Rudolph--Johnson定理，证明坐标函数在×2×3半群作用下的近返回或对本征函数的贴近会迫使分辨率熵变小或衰减，为Furstenberg猜想提供约束。
AI 中文摘要
在配套文章 \ref{RJnodyn} 中，本文作者证明了 Rudolph--Johnson 定理的一种在有限分辨率下有效的形式，该形式将 $\times 2 \times 3$ 不变测度的低频傅里叶系数以其归一化熵为界。此处我们从相反方向解读该界，将其视为对 Furstenberg $\times 2 \times 3$ 猜想潜在反例中本征函数行为的约束。我们证明，坐标函数 $x \to e^{2\pi ix}$ 在由 $2$ 和 $3$ 生成的半群作用下对自身的近返回迫使分辨率熵变小，而坐标函数对 $x \to 2x$ 的本征函数的贴近则迫使其衰减。这些速率在贴近速率和特征值的算术性质上是显式的。
英文摘要
In a companion article \cite{RJnodyn} the present authors proved a form of the Rudolph--Johnson theorem which is effective at finite resolution, bounding the low-frequency Fourier coefficients of a $\times 2 \times 3$-invariant measure by its normalized entropy. Here we read that bound in the other direction, as a constraint on eigenfunction behavior in potential counterexamples to Furstenberg's $\times 2 \times 3$ conjecture. We show that a near return of the coordinate function $x \mapsto e^{2πix}$ to itself under the semigroup generated by $2$ and $3$ forces the resolution entropy to be small, and that adherence of the coordinate function to eigenfunctions of $x \mapsto 2x$ forces it to decay. These rates are explicit in the rate of adherence and in the arithmetic of the eigenvalues.
