Furstenberg–Bergelson–Leibman pointwise convergence conjecture

Let $d,k,m\in\mathbb N$. Let $(X,\mathcal B(X),\mu)$ be a probability measure space, and let $T_1,\ldots,T_d:X\to X$ be invertible measure-preserving transformations generating a nilpotent group. Let $P_{1,1},\ldots,P_{d,m}\in\mathbb Z[\mathrm n_1,\ldots,\mathrm n_k]$ be $k$-variate polynomials with integer coefficients, and let $f_1,\ldots,f_m\in L^\infty(X)$. Define

$$A_{N_1,\ldots,N_k;X,T_1,\ldots,T_d}^{P_{1,1},\ldots,P_{d,m}}(f_1,\ldots,f_m)(x)=\mathbb E_{n\in[N_1]\times\cdots\times[N_k]}\prod_{j=1}^m f_j\bigl(T_1^{P_{1,j}(n)}\cdots T_d^{P_{d,j}(n)}x\bigr).$$

Furstenberg–Bergelson–Leibman conjecture. These averages converge for $\mu$-almost every $x\in X$ as $\min\\{N_1,\ldots,N_k\\}\to\infty$. When $N_1=\cdots=N_k=N$, write the averages as $A_{N;X,T_1,\ldots,T_d}^{P_{1,1},\ldots,P_{d,m}}$.

This is a central open problem in pointwise ergodic theory and modern Fourier analysis. The conjecture originated with Furstenberg and was later published by Bergelson and Leibman. Convergence can fail when the transformations generate a solvable non-nilpotent group, supporting the nilpotent hypothesis.