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

Abstract

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 $f\in L^r(X,\mu )$ , $g\in L^{r^{\prime }}(X,\mu )$ such that $\frac{1}{r}+\frac{1}{r^{\prime }}=1$, 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 $$\frac{1}{{\pi }_N}\sum _{\stackrel{p\le N}{p\text{ prime}}}f(T^{P(p)}x)g(T^{Q(p)}x),$$ converge. Here ${\pi }_N$ is the number of prime in $[1,N]$.

30 pages. Comments and questions on scientific matter are welcome

References in corpus (1)

• Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem

Cited by in corpus (1)

• On the polynomials homogeneous ergodic bilinear averages with Liouville and Möbius weights