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Pointwise convergence of ergodic averages with M\"obius weight

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arxiv 2401.03174 v2 pith:P5RD4R3U submitted 2024-01-06 math.DS math.NT

Joni Teräväinen This is my paper · ORCID
classification math.DSmath.NT
keywords averagesergodicpointwisealignconvergenceldotsmultiplepolynomial
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abstract

Let $(X,\nu,T)$ be a measure-preserving system, and let $P_1,\ldots, P_k$ be polynomials with integer coefficients. We prove that, for any $f_1,\ldots, f_k\in L^{\infty}(X)$, the M\"obius-weighted polynomial multiple ergodic averages \begin{align*}\frac{1}{N}\sum_{n\leq N}\mu(n)f_1(T^{P_1(n)}x)\cdots f_k(T^{P_k(n)}x) \end{align*} converge to $0$ pointwise almost everywhere. Specialising to $P_1(y)=y, P_2(y)=2y$, this solves a problem of Frantzikinakis. We also prove pointwise convergence for a more general class of multiplicative weights for multiple ergodic averages involving distinct degree polynomials. For the proofs we establish some quantitative generalised von Neumann theorems for polynomial configurations that are of independent interest.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Structure of 2-step nilpotent ergodic averages for distinct-degree polynomials

    math.DS 2026-07 conditional novelty 8.0 of 10

    For 2-step nilpotent actions, polynomial averages with distinct-degree iterates converge to the product of the integrals, and the joint ergodicity conjecture for Z^D polynomial actions is fully resolved.

  2. Pointwise convergence of polynomial multiple ergodic averages along the primes

    math.DS 2025-05 conditional novelty 8.0 of 10

    Prime-weighted polynomial multiple ergodic averages converge pointwise almost everywhere for arbitrary k and distinct-degree integer polynomials, with r-variational estimates for every r > 2.

  3. A Unified Approach to Two Pointwise Ergodic Theorems: Double Recurrence and Return Times

    math.DS 2025-01 conditional novelty 7.0 of 10

    For every real alpha and rational gamma, averaging with powers T^{floor(alpha n)} and T^{floor(gamma n)} converges almost everywhere, and the return-times analogue holds for aperiodic systems.

  4. On weighted multilinear polynomial averages in finite fields

    math.NT 2025-07 conditional novelty 6.0 of 10

    Weighted multilinear polynomial averages in finite fields are controlled by the u^(d+1)-norm of the weight, yielding quantitative convergence and new asymptotic formulas for multidimensional rational function progress...

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