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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. 

Abstract We prove uniform$$\ell ^2$$ ${\ell }^2$-valued maximal inequalities for polynomial ergodic averages and truncated singular operators of Cotlar type modeled over multidimensional subsets of primes. In the averages case, we combine this with earlier one-parameter oscillation estimates (Mehlhop and Słomian in Math Ann, 2023,https://doi.org/10.1007/s00208-023-02597-8) to prove corresponding multiparameter oscillation estimates. This provides a fuller quantitative description of the pointwise convergence of the mentioned averages and is a generalization of the polynomial Dunford–Zygmund ergodic theorem attributed to Bourgain (Mirek et al. in Rev Mat Iberoam 38:2249–2284, 2022). 

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. 

Abstract Let$$P_1, \ldots , P_m \in \mathbb {K}[\mathrm {y}]$$be polynomials with distinct degrees, no constant terms and coefficients in a general local field$$\mathbb {K}$$. We give a quantitative count of the number of polynomial progressions$$x, x+P_1(y), \ldots , x + P_m(y)$$lying in a set$$S\subseteq \mathbb {K}$$of positive density. The proof relies on a general$$L^{\infty }$$inverse theorem which is of independent interest. This inverse theorem implies a Sobolev improving estimate for multilinear polynomial averaging operators which in turn implies our quantitative estimate for polynomial progressions. This general Sobolev inequality has the potential to be applied in a number of problems in real, complex andp-adic analysis. 

Abstract We prove the uniform oscillation and jump inequalities for the polynomial ergodic averages modeled over multi-dimensional subsets of primes. This is a contribution to the Rosenblatt–Wierdl conjecture (Lond Math Soc Lect Notes 205:3–151, 1995, Problem 4.12, p. 80) with averages taken over primes. These inequalities provide endpoints for ther-variational estimates obtained by Trojan (Math Ann 374:1597–1656, 2019). 

Abstract We prove convergence in norm and pointwise almost everywhere on$$L^p$$,$$p\in (1,\infty )$$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow–Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages.