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We establish characteristic factors for natural classes of polynomial multiple ergodic averages in rings of integers and derive corresponding Khintchine-type recurrence theorems, extending results of Frantzikinakis and Kra and of Frantzikinakis about polynomial configurations in $mathbb{Z}$. Using previous work of Griesmer and of the second author and Robertson, we reduce the problem of finding characteristic factors to proving a result on equidistribution of polynomial orbits in nilmanifolds, which is of independent interest.
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The purpose of this paper is to study the phenomenon of large intersections in the framework of multiple recurrence for measure-preserving actions of countable abelian groups. Among other things, we show: (1) If $G$ is a countable abelian group and $varphi, psi : G to G$ are homomorphisms such that $varphi(G)$, $psi(G)$, and $(psi - varphi)(G)$ have finite index in $G$, then for every ergodic measure-preserving system $(X, mathcal{B}, mu, (T_g)_{g in G})$, every set $A in mathcal{B}$, and every $varepsilon > 0$, the set ${g in G : mu(A cap T_{varphi(g)}^{-1}A cap T_{psi(g)}^{-1}A) > mu(A)^3 - varepsilon}$ is syndetic. (2) If $G$ is a countable abelian group and $r,s in mathbb{Z}$ are integers such that $rG$, $sG$, and $(r pm s)G$ have finite index in $G$, then for every ergodic measure-preserving system $(X, mathcal{B}, mu, (T_g)_{g in G})$, every set $A in mathcal{B}$, and every $varepsilon > 0$, the set ${g in G : mu(A cap T_{rg}^{-1}A cap T_{sg}^{-1}A cap T_{(r+s)g}^{-1}A) > mu(A)^4 - varepsilon}$ is syndetic. In particular, these extend and generalize results of Bergelson, Host, and Kra concerning $mathbb{Z}$-actions and of Bergelson, Tao, and Ziegler concerning $mathbb{F}_p^{infty}$-actions. Using an ergodic version of the Furstenberg correspondence principle, we obtain new combinatorial applications. We also discuss numerous examples shedding light on the necessity of the various hypotheses above. Our results lead to a number of interesting questions and conjectures, formulated in the introduction and at the end of the paper.
We establish new recurrence and multiple recurrence results for a rather large family $mathcal{F}$ of non-polynomial functions which includes tempered functions defined in [11], as well as functions from a Hardy field with the property that for some $ellin mathbb{N}cup{0}$, $lim_{xtoinfty }f^{(ell)}(x)=pminfty$ and $lim_{xtoinfty }f^{(ell+1)}(x)=0$. Among other things, we show that for any $finmathcal{F}$, any invertible probability measure preserving system $(X,mathcal{B},mu,T)$, any $Ainmathcal{B}$ with $mu(A)>0$, and any $epsilon>0$, the sets of returns $$ R_{epsilon, A}= big{ninmathbb{N}:mu(Acap T^{-lfloor f(n)rfloor}A)>mu^2(A)-epsilonbig} $$ and $$ R^{(k)}_{A}= big{ ninmathbb{N}: mubig(Acap T^{lfloor f(n)rfloor}Acap T^{lfloor f(n+1)rfloor}Acapcdotscap T^{lfloor f(n+k)rfloor}Abig)>0big} $$ possess somewhat unexpected properties of largeness; in particular, they are thick, i.e., contain arbitrarily long intervals.
We provide various counter examples for quantitative multiple recurrence problems for systems with more than one transformation. We show that $bullet$ There exists an ergodic system $(X,mathcal{X},mu,T_1,T_2)$ with two commuting transformations such that for every $0<ell< 4$, there exists $Ainmathcal{X}$ such that $$mu(Acap T_{1}^{-n}Acap T_{2}^{-n}A)<mu(A)^{ell} text{ for every } n eq 0;$$ $bullet$ There exists an ergodic system $(X,mathcal{X},mu,T_1,T_2, T_{3})$ with three commuting transformations such that for every $ell>0$, there exists $Ainmathcal{X}$ such that $$mu(Acap T_{1}^{-n}Acap T_{2}^{-n}Acap T_{3}^{-n}A)<mu(A)^{ell} text{ for every } n eq 0;$$ $bullet$ There exists an ergodic system $(X,mathcal{X},mu,T_1,T_2)$ with two transformations generating a 2-step nilpotent group such that for every $ell>0$, there exists $Ainmathcal{X}$ such that $$mu(Acap T_{1}^{-n}Acap T_{2}^{-n}A)<mu(A)^{ell} text{ for every } n eq 0.$$
Let $(X, mathcal{B},mu,T)$ be an ergodic measure preserving system, $A in mathcal{B}$ and $epsilon>0$. We study the largeness of sets of the form begin{equation*} begin{split} S = left{ ninmathbb{N}colonmu(Acap T^{-f_1(n)}Acap T^{-f_2(n)}Acapldotscap T^{-f_k(n)}A)> mu(A)^{k+1} - epsilon right} end{split} end{equation*} for various families ${f_1,dots,f_k}$ of sequences $f_icolon mathbb{N} to mathbb{N}$. For $k leq 3$ and $f_{i}(n)=i f(n)$, we show that $S$ has positive density if $f(n)=q(p_n)$ where $q in mathbb{Z}[x]$ satisfies $q(1)$ or $q(-1) =0$ and $p_n$ denotes the $n$-th prime; or when $f$ is a certain Hardy field sequence. If $T^q$ is ergodic for some $q in mathbb{N}$, then for all $r in mathbb{Z}$, $S$ is syndetic if $f(n) = qn + r$. For $f_{i}(n)=a_{i}n$, where $a_{i}$ are distinct integers, we show that $S$ can be empty for $kgeq 4$, and for $k = 3$ we found an interesting relation between the largeness of $S$ and the abundance of solutions to certain linear equations in sparse sets of integers. We also provide some partial results when the $f_{i}$ are distinct polynomials.
The Furstenberg-Sarkozy theorem asserts that the difference set $E-E$ of a subset $E subset mathbb{N}$ with positive upper density intersects the image set of any polynomial $P in mathbb{Z}[n]$ for which $P(0)=0$. Furstenbergs approach relies on a correspondence principle and a polynomial version of the Poincare recurrence theorem, which is derived from the ergodic-theoretic result that for any measure-preserving system $(X,mathcal{B},mu,T)$ and set $A in mathcal{B}$ with $mu(A) > 0$, one has $c(A):= lim_{N to infty} frac{1}{N} sum_{n=1}^N mu(A cap T^{-P(n)}A) > 0.$ The limit $c(A)$ will have its optimal value of $mu(A)^2$ when $T$ is totally ergodic. Motivated by the possibility of new combinatorial applications, we define the notion of asymptotic total ergodicity in the setting of modular rings $mathbb{Z}/Nmathbb{Z}$. We show that a sequence of modular rings $mathbb{Z}/N_mmathbb{Z}$, $m in mathbb{N},$ is asymptotically totally ergodic if and only if $mathrm{lpf}(N_m)$, the least prime factor of $N_m$, grows to infinity. From this fact, we derive some combinatorial consequences, for example the following. Fix $delta in (0,1]$ and a (not necessarily intersective) polynomial $Q in mathbb{Q}[n]$ such that $Q(mathbb{Z}) subseteq mathbb{Z}$, and write $S = { Q(n) : n in mathbb{Z}/Nmathbb{Z}}$. For any integer $N > 1$ with $mathrm{lpf}(N)$ sufficiently large, if $A$ and $B$ are subsets of $mathbb{Z}/Nmathbb{Z}$ such that $|A||B| geq delta N^2$, then $mathbb{Z}/Nmathbb{Z} = A + B + S$.
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