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Abstract We establish pointwise convergence for nonconventional ergodic averages taken along $$\lfloor p^{c}\rfloor $$, where $$p$$ is a prime number and $$c\in (1,4/3)$$ on $$L^{r}$$, $$r\in (1,\infty )$$. In fact, we consider averages along more general sequences $$\lfloor h(p)\rfloor $$, where $$h$$ belongs in a wide class of functions, the so-called $$c$$-regularly varying functions. We also establish uniform multiparameter oscillation estimates for our ergodic averages and the corresponding multiparameter pointwise ergodic theorem in the spirit of Dunford and Zygmund. A key ingredient of our approach are certain exponential sum estimates, which we also use for establishing a Waring-type result. Assuming that the Riemann zeta function has any zero-free strip upgrades our exponential sum estimates to polynomially saving ones and this makes a conditional result regarding the behavior of our ergodic averages on $$L^{1}$$ to not seem entirely out of reach. 

We introduce adversarial learning methods for data-driven generative modeling of dynamics of n-th-order stochastic systems. Our approach builds on Generative Adversarial Networks (GANs) with generative model classes based on stable m-step stochastic numerical integrators. From observations of trajectory samples, we introduce methods for learning long-time predictors and stable representations of the dynamics. Our approaches use discriminators based on Maximum Mean Discrepancy (MMD), training protocols using both conditional and marginal distributions, and methods for learning dynamic responses over different time-scales. We show how our approaches can be used for modeling physical systems to learn force-laws, damping coefficients, and noise-related parameters. Our adversarial learning approaches provide methods for obtaining stable generative models for dynamic tasks including long-time prediction and developing simulations for stochastic systems. 

Abstract We establish weak-type (1, 1) bounds for the maximal function associated with ergodic averaging operators modeled on a wide class of thin deterministic setsB. As a corollary we obtain the corresponding pointwise convergence result on$$L^1$$ $L^1$. This contributes yet another counterexample for the conjecture of Rosenblatt and Wierdl from 1991 asserting the failure of pointwise convergence on$$L^1$$ $L^1$of ergodic averages along arithmetic sets with zero Banach density. The second main result is a multiparameter pointwise ergodic theorem in the spirit of Dunford and Zygmund alongBon$$L^p$$ $L^p$,$$p>1$$ $p>1$, which is derived by establishing uniform oscillation estimates and certain vector-valued maximal estimates.