A Criterion for Equidistribution along the $Ω$ Function over Polynomial Sequences with Applications

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 $\Omega (|P (n_1, ..., n_d)|)$. 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 $\mu$, 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^{ \Omega (|P (n_1, n_2)| ) } x \big) = \int_{X} f \ \mathrm{d} \mu . \end{equation*} Moreover, we prove in the appendix a related conjecture of C\'espedes and Donoso over number fields.