论文标题
沿着强壮领域的函数沿函数的多个千古平均值：收敛，复发和组合应用
Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications
论文作者
论文摘要
我们获得了有关沿着强壮场的函数的多个沿着函数的收敛和复发的新结果。除其他外，我们确认了弗朗西基纳基斯在[fra10中提出的一些猜想； fra16]并获得包含特殊案例的组合应用，这些案例是Szemeredi定理的几种先前已知的（多项式和非多项式）扩展[bl96; bll08; FW09; fra10; BMR17]。我们的结果的新型特征之一是在先前的工作中不存在的，即它们允许多项式和非多项式函数的混合物。作为插图，假设$ f_i（t）= a_ {i，1} t^{c_ {c_ {i，1}}+\ cdots+a_ {i，d} t^{c_ {c_ {i，d}} $ for $ c_ {i，j}> 0 $ and $ a_____ {然后 $ \ bullet $对于任何度量保留系统$（x，{\ mathcal b}，μ，t）$和$ h_1，\ dots，h_k \ in l^\ infty（x）$，limit $ \ lim_ {n \ n \ to \ infty} t^{[f_1（n）]} h_1 \ cdots t^{[f_k（n）]} h_k $$存在于$ l^2 $中; 任何$ e \ subset \ mathbb {n} $带有$ \ overline {\ mathrm {d}}（e）> 0 $的$ \ bullet $，有$ a，n \ in \ mathbb {n} $，因此$ a，n \ in $ ar E $。 我们还表明，如果$ f_1，\ dots，f_k $属于一个强大的领域，具有多项式增长，并且它们的线性组合没有线性组合是多项式的，那么对于任何度量保留系统$（x，x，{\ nathcal b}，μ $ \ limsup_ {n \ to \ infty} \ frac {1} {n} \ sum_ {n = 1}^nμ\ big（a \ cap t^{ - [f_1（n）] t^{ - [f_k（n）]} a \ big）\，\ geq \，μ（a）^{k+1}。$$
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 $A\in{\mathcal B}$, $$\limsup_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nμ\Big(A\cap T^{-[ f_1(n) ]}A\cap\ldots\cap T^{-[f_k(n)]}A\Big)\,\geq\,μ(A)^{k+1}.$$
