On the homogeneous ergodic bilinear averages with $1$-bounded multiplicative weights
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866917901022265344 |
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| author | Abdalaoui, el Houcein el |
| author_facet | Abdalaoui, el Houcein el |
| contents | We establish a generalization of Bourgain double recurrence theorem and ergodic Bourgain-Sarnak's theorem by proving that for any aperiodic $1$-bounded multiplicative function $\boldsymbolν$, for any map $T$ acting on a probability space $(X,\mathcal{A},μ)$, for any integers $a,b$, for any $f,g \in L^2(X)$, and for almost all $x \in X$, we have \[\frac{1}{N} \sum_{n=1}^{N} \boldsymbolν(n) f(T^{a n}x)g(T^{bn}x) \xrightarrow[N\rightarrow +\infty]{} 0.\] We further present with proof the key ingredients of Bourgain's proof of his double recurrence theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_06323 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the homogeneous ergodic bilinear averages with $1$-bounded multiplicative weights Abdalaoui, el Houcein el Dynamical Systems Number Theory Primary: 37A30, Secondary: 28D05, 5D10, 11B30, 11N37, 37A45 We establish a generalization of Bourgain double recurrence theorem and ergodic Bourgain-Sarnak's theorem by proving that for any aperiodic $1$-bounded multiplicative function $\boldsymbolν$, for any map $T$ acting on a probability space $(X,\mathcal{A},μ)$, for any integers $a,b$, for any $f,g \in L^2(X)$, and for almost all $x \in X$, we have \[\frac{1}{N} \sum_{n=1}^{N} \boldsymbolν(n) f(T^{a n}x)g(T^{bn}x) \xrightarrow[N\rightarrow +\infty]{} 0.\] We further present with proof the key ingredients of Bourgain's proof of his double recurrence theorem. |
| title | On the homogeneous ergodic bilinear averages with $1$-bounded multiplicative weights |
| topic | Dynamical Systems Number Theory Primary: 37A30, Secondary: 28D05, 5D10, 11B30, 11N37, 37A45 |
| url | https://arxiv.org/abs/2012.06323 |