AI 中文总结
本文证明仅凭熵条件即可使圆周上的测度在亚空隙乘子自同态下趋于 Lebesgue 测度，给出有限熵界并推广 Rudolph-Johnson 定理，同时揭示 Lyons 猜想的成立与失败条件。
AI 中文摘要
我们证明，仅凭熵，在没有任何不变性假设的情况下，就能迫使圆周上的 Borel 概率测度在每一组亚空隙的乘法自同态 $x \mapsto mx$ 作用下趋向于 Lebesgue 测度。此处，我们将无穷乘子集合 $m$ 称为亚空隙的，如果其连续比值趋于 $1$。此类集合的例子包括完全立方数之后的素数、配分函数的取值范围、整数 $\lfloor n^{\log n} \rfloor$ 以及由 $2$ 和 $3$ 生成的半群，后者是 Furstenberg 猜想的经典情形。具体地，我们证明每个测度 $\mu$ 都有一个自同态的弱星极限，该极限控制着按 $\mu$ 的上熵维数缩放的 Lebesgue 测度。这些定理在有限熵分辨率下是有效的，并且我们为素数和完全幂提供了相应的界。半群情形可以解释为 Rudolph-Johnson 定理的一种无不变性形式。我们还证明，对于每个亚空隙乘子集合，在全熵维数下 Lyons 猜想的一个类比成立，而对于 $3$ 的幂构成的空隙半群，该类比则以任意大的因子失败。证明将傅里叶分析与乘法自同态的代数结构相结合。
英文摘要
We prove that entropy alone, with no invariance hypothesis, forces a Borel probability measure on the circle toward Lebesgue measure under every sublacunary set of multiplicative endomorphisms $x \mapsto mx$. Here, we refer to an infinite set of multipliers $m$ as sublacunary if its consecutive ratios tend to $1$. Examples of such sets include the primes following perfect cubes, the range of the partition function, the integers $\lfloor n^{\log n} \rfloor$ and the semigroup generated by $2$ and $3$ which is the classical case of Furstenberg's conjecture. Specifically, we show that every measure $μ$ has a weak-star limit of endomorphs which dominates Lebesgue measure scaled by the upper entropy dimension of $μ$. These theorems are effective at finite entropy resolutions, and we provide the associated bounds for the primes and the perfect powers. The semigroup case may be interpreted as an invariance-free form of the Rudolph-Johnson theorem. We also show that an analog of Lyons' conjecture at full entropy dimension for every sublacunary multiplier set while failing by an arbitrarily large factor for the lacunary semigroup of powers of $3$. The proofs combine Fourier analysis with the algebraic structure of the multiplicative endomorphisms.
Comments87 pages
