关于多项式序列上沿Ω函数的等分布准则及其应用
A Criterion for Equidistribution along the $Ω$ Function over Polynomial Sequences with Applications
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中文总结 AI 辅助
研究关于整系数齐次多项式\(P\)，建立沿\(\Omega(|P(n_1,...,n_d)|)\)的等分布准则，证明了相关定理变体，并在附录中证明了关于数域的相关猜想。
中文摘要 AI 辅助
设\(P(Y_1,...,Y_d)\)是一个固定的整系数齐次多项式。本文建立了沿\(\Omega(|P(n_1,...,n_d)|)\)的遍历平均的定量等分布准则。通过Lachand的估计，证明了Bergelson和Richter定理的一个变体：若\(P\)是不可约二元三次形式且\((X,T)\)是具有唯一不变测度\(\mu\)的唯一遍历系统，则对任意\(x\in X\)和\(f\in C(X)\)，有相应极限等式成立。此外，在附录中证明了Céspedes和Donoso关于数域的一个相关猜想。
英文摘要
Let $P (Y_1, ..., Y_d)$ be a certain fixed homogeneous polynomial of integral coefficients. In this paper, we establish a quantitative equidistribution criterion for the ergodic averages along $Ω(|P (n_1, ..., n_d)|)$. Consequently, by an estimate of Lachand, we prove the following variant of a theorem of Bergelson and Richter: if $P$ is an irreducible binary cubic form and $ (X, T)$ is a uniquely ergodic system with unique invariant measure $μ$, then for any $x \in X$ and $f \in C(X)$, \begin{equation*} \lim_{N \rightarrow \infty} \frac 1 {N^2} {\mathop{\sum\sum}_{n_1, n_2 \leqslant N}} f \big( T^{ Ω(|P (n_1, n_2)| ) } x \big) = \int_{X} f \ \mathrm{d} μ. \end{equation*} Moreover, we prove in the appendix a related conjecture of Céspedes and Donoso over number fields.
