Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications
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| Formato: | Preprint |
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2020
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| _version_ | 1866915781669814272 |
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| author | Bergelson, Vitaly Moreira, Joel Richter, Florian K. |
| author_facet | Bergelson, Vitaly Moreira, Joel Richter, Florian K. |
| contents | We obtain new results pertaining to convergence and recurrence of multiple ergodic averages along functions from a Hardy field. Among other things, we confirm some of the conjectures posed by Frantzikinakis in [Fra10; Fra16] and obtain combinatorial applications which contain, as rather special cases, several previously known (polynomial and non-polynomial) extensions of Szemeredi's theorem on arithmetic progressions [BL96; BLL08; FW09; Fra10; BMR17]. One of the novel features of our results, which is not present in previous work, is that they allow for a mixture of polynomials and non-polynomial functions. As an illustration, assume $f_i(t)=a_{i,1}t^{c_{i,1}}+\cdots+a_{i,d}t^{c_{i,d}}$ for $c_{i,j}>0$ and $a_{i,j}\in\mathbb{R}$. Then
$\bullet$ for any measure preserving system $(X,\mathcal{B},μ,T)$ and $h_1,\dots,h_k\in L^\infty(X)$, the limit $$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N T^{[f_1(n)]}h_1\cdots T^{[f_k(n)]}h_k$$ exists in $L^2$;
$\bullet$ for any $E\subset \mathbb{N}$ with $\overline{\mathrm{d}}(E)>0$ there are $a,n\in\mathbb{N}$ such that $\{a,\, a+[f_1(n)],\ldots,a+[f_k(n)]\}\subset E$.
We also show that if $f_1,\dots,f_k$ belong to a Hardy field, have polynomial growth, and are such that no linear combination of them is a polynomial, then for any measure preserving system $(X,{\mathcal B},μ,T)$ and any $A\in{\mathcal B}$, $$\limsup_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nμ\Big(A\cap T^{-[ f_1(n) ]}A\cap\ldots\cap T^{-[f_k(n)]}A\Big)\,\geq\,μ(A)^{k+1}.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2006_03558 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications Bergelson, Vitaly Moreira, Joel Richter, Florian K. Dynamical Systems Combinatorics 37A44, 28D05, 05D10, 11B30 We obtain new results pertaining to convergence and recurrence of multiple ergodic averages along functions from a Hardy field. Among other things, we confirm some of the conjectures posed by Frantzikinakis in [Fra10; Fra16] and obtain combinatorial applications which contain, as rather special cases, several previously known (polynomial and non-polynomial) extensions of Szemeredi's theorem on arithmetic progressions [BL96; BLL08; FW09; Fra10; BMR17]. One of the novel features of our results, which is not present in previous work, is that they allow for a mixture of polynomials and non-polynomial functions. As an illustration, assume $f_i(t)=a_{i,1}t^{c_{i,1}}+\cdots+a_{i,d}t^{c_{i,d}}$ for $c_{i,j}>0$ and $a_{i,j}\in\mathbb{R}$. Then $\bullet$ for any measure preserving system $(X,\mathcal{B},μ,T)$ and $h_1,\dots,h_k\in L^\infty(X)$, the limit $$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N T^{[f_1(n)]}h_1\cdots T^{[f_k(n)]}h_k$$ exists in $L^2$; $\bullet$ for any $E\subset \mathbb{N}$ with $\overline{\mathrm{d}}(E)>0$ there are $a,n\in\mathbb{N}$ such that $\{a,\, a+[f_1(n)],\ldots,a+[f_k(n)]\}\subset E$. We also show that if $f_1,\dots,f_k$ belong to a Hardy field, have polynomial growth, and are such that no linear combination of them is a polynomial, then for any measure preserving system $(X,{\mathcal B},μ,T)$ and any $A\in{\mathcal B}$, $$\limsup_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nμ\Big(A\cap T^{-[ f_1(n) ]}A\cap\ldots\cap T^{-[f_k(n)]}A\Big)\,\geq\,μ(A)^{k+1}.$$ |
| title | Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications |
| topic | Dynamical Systems Combinatorics 37A44, 28D05, 05D10, 11B30 |
| url | https://arxiv.org/abs/2006.03558 |