Pointwise convergence of ergodic averages with Möbius weight
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910553576833024 |
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| author | Teräväinen, Joni |
| author_facet | Teräväinen, Joni |
| contents | Let $(X,ν,T)$ 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 Möbius-weighted polynomial multiple ergodic averages \begin{align*}\frac{1}{N}\sum_{n\leq N}μ(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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03174 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pointwise convergence of ergodic averages with Möbius weight Teräväinen, Joni Dynamical Systems Number Theory 37A44, 37A30, 11B30 Let $(X,ν,T)$ 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 Möbius-weighted polynomial multiple ergodic averages \begin{align*}\frac{1}{N}\sum_{n\leq N}μ(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. |
| title | Pointwise convergence of ergodic averages with Möbius weight |
| topic | Dynamical Systems Number Theory 37A44, 37A30, 11B30 |
| url | https://arxiv.org/abs/2401.03174 |