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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.
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Lu, Guozhen (Ed.)Abstract We discuss some of our work on averages along polynomial sequences in nilpotent groups of step 2. Our main results include boundedness of associated maximal functions and singular integrals operators, an almost everywhere pointwise convergence theorem for ergodic averages along polynomial sequences, and a nilpotent Waring theorem. Our proofs are based on analytical tools, such as a nilpotent Weyl inequality, and on complex almost-orthogonality arguments that are designed to replace Fourier transform tools, which are not available in the noncommutative nilpotent setting. In particular, we present what we call anilpotent circle methodthat allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.more » « less
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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.more » « less
