Simple proof of Bourgain bilinear ergodic theorem and its extension to polynomials and polynomials in primes

We first present a modern simple proof of the classical ergodic Birkhoff's theorem and Bourgain's homogeneous bilinear ergodic theorem. This proof used the simple fact that the shift map on integers has a simple Lebesgue spectrum. As a consequence, we establish that the homogeneous bilinear ergodic averages along polynomials and polynomials in primes converge almost everywhere, that is, for any invertible measure preserving transformation \(T\), acting on a probability space \((X, \mathcal{B}, \mu)\), for any \(f \in L^r(X,\mu)\) , \(g \in L^{r'}(X,\mu)\) such that \(\frac{1}{r}+\frac{1}{r'}= 1\), for any non-constant polynomials \(P(n),Q(n), n \in \mathbb{Z}\), taking integer values, and for almost all \(x \in X\), we have, $$\frac{1}{N}\sum_{n=1}^{N}f(T^{P(n)}x) g(T^{Q(n)}x),$$ and $$\frac{1}{\pi_N}\sum_{\overset{p \leq N}{p\textrm{~~prime}}}f(T^{P(p)}x) g(T^{Q(p)}x),$$ converge. Here \(\pi_N\) is the number of prime in \([1,N]\).