Pointwise ergodic theorems for non-conventional bilinear averages along $(\lfloor n^c\rfloor,-\lfloor n^c\rfloor)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910861229031424 |
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| author | Daskalakis, Leonidas |
| author_facet | Daskalakis, Leonidas |
| contents | For every $c\in(1,23/22)$ and every probability dynamical system $(X,\mathcal{B},μ,T)$ we prove that for any $f,g\in L^{\infty}_μ(X)$ the bilinear ergodic averages \[ \frac{1}{N}\sum_{n=1}^Nf(T^{\lfloor n^c\rfloor}x)g(T^{-\lfloor n^c\rfloor}x)\quad\text{converge for $μ$-a.e. $x\in X$.} \] In fact, we consider more general sparse orbits $(\lfloor h(n)\rfloor,-\lfloor h(n)\rfloor)_{n\in\mathbb{N}}$, where $h$ belongs to the class of the so-called $c$-regularly varying functions. This is the first pointwise result for bilinear ergodic averages taken along deterministic sparse orbits where modulation invariance is present. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03976 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pointwise ergodic theorems for non-conventional bilinear averages along $(\lfloor n^c\rfloor,-\lfloor n^c\rfloor)$ Daskalakis, Leonidas Dynamical Systems Classical Analysis and ODEs For every $c\in(1,23/22)$ and every probability dynamical system $(X,\mathcal{B},μ,T)$ we prove that for any $f,g\in L^{\infty}_μ(X)$ the bilinear ergodic averages \[ \frac{1}{N}\sum_{n=1}^Nf(T^{\lfloor n^c\rfloor}x)g(T^{-\lfloor n^c\rfloor}x)\quad\text{converge for $μ$-a.e. $x\in X$.} \] In fact, we consider more general sparse orbits $(\lfloor h(n)\rfloor,-\lfloor h(n)\rfloor)_{n\in\mathbb{N}}$, where $h$ belongs to the class of the so-called $c$-regularly varying functions. This is the first pointwise result for bilinear ergodic averages taken along deterministic sparse orbits where modulation invariance is present. |
| title | Pointwise ergodic theorems for non-conventional bilinear averages along $(\lfloor n^c\rfloor,-\lfloor n^c\rfloor)$ |
| topic | Dynamical Systems Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2503.03976 |