发表机构
Institute of Mathematics, Polish Academy of Sciences; Institute of Mathematics, University of Wrocław; Department of Mathematics, Rutgers University; Department of Mathematics, University of Memphis（波兰科学院数学研究所; 弗罗茨瓦夫大学数学研究所; 罗格斯大学数学系; 孟菲斯大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文首次证明沿中间增长序列（如 $\lfloor \exp((\log n)^c)\rfloor$，$c\in(1,8/7)$）的遍历平均在 $L^p$ 中普适收敛，解决了1980年代以来的开放问题，方法结合单频圆法与Vinogradov指数和估计。
AI 中文摘要
我们建立了第一个关于沿显式且确定的中间增长序列（即增长速度超过任何多项式但慢于任何指数）的遍历平均的点态收敛结果。特别地，我们证明序列 $\lfloor \exp((\log n)^c)\rfloor_{n\in\mathbb{Z}_+}$（其中 $c\in(1,8/7)$）对每个 $p\in(1,\infty]$ 都是普适的 $L^p$-好序列。这肯定地回答了一个可追溯至1980年代中期的开放问题，并推动了Bellow在1980年代早期发起的关于点态遍历定理中 $L^p$-好序列刻画的研究纲领。证明结合了所谓单频圆法与Vinogradov方法的精细应用，用于估计相位涉及 $\lfloor \exp((\log n)^c)\rfloor_{n\in\mathbb{Z}_+}$ 的指数和。我们分析的一个有趣特征——令人联想到研究Riemann zeta函数零区域时产生的估计——是论证依赖于经典Vinogradov方法，即需要估计Vinogradov丢番图方程组解的数量，并显式依赖于该系统的参数。
英文摘要
We establish the first pointwise convergence result for ergodic averages with iterates along explicit and deterministic sequences of intermediate growth, that is, growing faster than any polynomial but slower than any exponential. In particular, we show that the sequence $(\lfloor \exp((\log n)^c)\rfloor)_{n\in\mathbb{Z}+}$, with $c\in(1,8/7)$, is universally $L^p$-good for every $p\in(1,\infty]$. This gives an affirmative answer to an open problem dating back to the mid 1980s and contributes to Bellow's program, initiated in the earlier part of the same decade, on the characterization of $L^p$-good sequences in pointwise ergodic theorems. The proof combines the so-called one-frequency circle method with a delicate application of Vinogradov's method for estimating exponential sums whose phases involve $\big(\lfloor \exp((\log n)^c)\rfloor\big)_{n\in\mathbb{Z}_+}$. An interesting feature of our analysis, reminiscent of estimates arising in the study of the zero-free region of the Riemann zeta function, is that the argument relies on the classical Vinogradov method, in the sense that it necessitates estimates on the number of solutions for the Vinogradov system of Diophantine equations with explicit dependence on the system's parameters.
Comments28 pages, no figures
