Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications

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 $(X,\mathcal{B},\mu ,T)$ and $h_1,\dots ,h_k\in L^{\mathrm{\infty }}(X)$, the limit $$\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N}\sum _{n=1}^N{T}^{[f_1(n)]}{h}_1\cdots T^{[f_k(n)]}{h}_k$$ exists in $L^2$; $\bullet$ for any $E\subset \mathbb{N}$ with $\bar{\mathrm{d}}(E)>0$ there are $a,n\in \mathbb{N}$ such that $\{a,a+[f_1(n)],\dots ,a+[f_k(n)]\}\subset E$. 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 $(X,\mathcal{B},\mu ,T)$ and any $A\in \mathcal{B}$, $$\underset{N\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{1}{N}\sum _{n=1}^N\mu (A\cap T^{-[f_1(n)]}A\cap \dots \cap T^{-[f_k(n)]}A)\ge \mu (A{)}^{k+1}\mathrm{.}$$