短区间上的多项式遍历平均
Polynomial Ergodic Averages Along Short Intervals
浏览论文内容
中文总结 AI 辅助
该研究证明了短区间上次数$d\geq2$、起始时间双间隙的多项式轨道在$1<p<\infty$且$c>(d-1)/d$时的$L^p$变分估计及几乎处处收敛，还指出端点$L^1$不成立，且密集起始时间序列具强扫出性质。
中文摘要 AI 辅助
我们研究左端点趋于无穷的短区间上多项式遍历平均的逐点收敛性。对于次数$d\geq2$且起始时间双间隙的多项式轨道，我们证明了$1<p<\infty$且$c>(d-1)/d$时的$L^p$变分估计，从而得到几乎处处收敛，这是短区间上多项式轨道的首个逐点遍历定理。我们还证明端点$L^1$在每个无穷子序列上不成立。此外，我们证明起始时间的密集序列具有强扫出性质。
英文摘要
We study pointwise convergence of polynomial ergodic averages over short intervals whose left endpoints tend to infinity. For a polynomial orbit of degree $d\geq2$ and doubly lacunary starting times, we prove $L^p$ variational estimates, and hence almost-everywhere convergence, for $1<p<\infty$ in the range $c>(d-1)/d$. This gives the first pointwise ergodic theorem for polynomial orbits along short intervals. We also show that the endpoint $L^1$ fails along every infinite subsequence. In a different direction, we prove that substantially denser sequences of starting times exhibit the strong sweeping-out property.
