Pointwise ergodic theorems for non-conventional bilinear polynomial averages

We establish convergence in norm and pointwise almost everywhere for the non-conventional (in sense of Furstenberg) bilinear polynomial ergodic averages $$ A_N(f,g)(x) := \frac{1}{N} \sum_{n=1}^{N} f(T^nx) g(T^{P(n)}x) as $N \to \infty$, where $T\colon X\to X$ is a measure-preserving transformation $\sigma$-finite measure space $(X,\mu), P(\mathrm{n})\in \mathbb{Z}[\mathrm{n}]$ degree $d \geq 2$, $f \in L^{p_1}(X)$, $g L^{p_2}(X)$ some $p_1, p_2 > 1$ with $\frac{1}{p_1} + \frac{1}{p_2} \leq 1$. also an $r$-variational inequality these (at lacunary scales) optimal range $r 2$. are able to ``break duality" by handling ranges exponents $p_1$, $p_2$ 1$, at cost increasing $r$ slightly. This gives affirmative answer Problem 11 from Frantzikinakis' open problems survey Furstenberg--Weiss (with $P(\mathrm{n})=\mathrm{n}^2)$, which variant Question 9 considered Bergelson his on Ergodic Ramsey Theory 1996. contribution Furstenberg--Bergelson--Leibman conjecture. Our methods combine techniques harmonic analysis recent inverse theorems Peluse Prendiville additive combinatorics. At large scales, adelic integers $\mathbb{A}_{\mathbb{Z}}$ plays role.