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

Vitaly Bergelson, Florian Richter and I have recently uploaded to the arXiv our new paper entitled “Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications”. We establish a new multiple recurrence result (and consequentially a new generalization of Szemerédi’s theorem) which contains several previously known results as special cases and many new results.

An example of a result that follows as a special case from our main theorem but was currently unknown is that for any positive real numbers ${a,b>0}$, any set ${E\subset{\mathbb N}}$ with positive upper density, i.e.

$\displaystyle \bar d(E):=\limsup_{N\rightarrow\infty}\frac{|E\cap\{1,\dots,N\}|}N>0,$

contains a configuration ${\{x,x+\lfloor n^a\rfloor,x_\lfloor n^b\rfloor\}}$ for some ${x,n\in{\mathbb N}}$. Here, as usual, we denote by ${\lfloor\cdot\rfloor}$ the floor function. This result was known when both ${a}$ and ${b}$ are integers, and when both ${a}$ and ${b}$ are non-integers, but the mixed case was, somewhat surprisingly, still unknown. Below the fold I briefly mention the history of related results and present the main steps in the proof, highlighting the main novelties.

— 1. History —

Szemerédi’s theorem, proved in 1975, states that any set ${E\subset{\mathbb N}}$ with positive upper density contains arbitrarily long finite arithmetic progressions. In symbols this means that ${\forall k\in{\mathbb N}}$ there exists ${x,n\in{\mathbb N}}$ such that ${\{x,x+n,x+2n,\dots,x+kn\}\subset E}$. An interesting question is to understand if further restrictions can be placed on ${n}$ or, more generally, for which functions ${f_1,\dots,f_k:{\mathbb N}\rightarrow{\mathbb N}}$ is it true that any set ${E\subset{\mathbb N}}$ with ${\bar d(E)>0}$ contain a configuration of the form ${\{x,x+f_1(n),\dots,x+f_k(n)\}}$?

Soon after Szemerédi’s proof, a different, ergodic theoretic, proof was found by Furstenberg, using a structure theory of measure preserving systems. This proof was more amenable to generalizations, and indeed it was using a similar ergodic theoretic approach that in 1996 Bergelson and Leibman proved a “polynomial Szemerédi theorem”. More precisely, in the one dimensional case, their result states that, whenever ${f_i\in{\mathbb Z}[x]}$ has ${f_i(0)=0}$ for ${i=1,\dots,k}$ and ${E\subset{\mathbb N}}$ has ${\bar d(E)>0}$, there are ${x,n\in{\mathbb N}}$ such that

$\displaystyle \big\{x,x+f_1(n),\dots,x+f_k(n)\big\}\subset E. \ \ \ \ \ (1)$

Taking ${f_i(n)=in}$ one recovers Szemerédi’s theorem from the Bergelson-Leibman theorem, and taking ${f_i(n)=in^2}$ one obtains the generalization that any set ${E\subset{\mathbb N}}$ with positive upper density contains arbitrarily long arithmetic progressions whose common difference is a perfect square.

Then in 2005 a stronger structure theory of measure preserving systems was discovered by Host and Kra (and independently by Ziegler soon after). This new structure theory allowed for several new developments. In 2008 Bergelson, Leibman and Lesigne extended the polynomial Szemerédi theorem to the following

Theorem 1 Let ${f_1,\dots,f_k\in{\mathbb Z}[x]}$. The following are equivalent:
• For every ${E\subset{\mathbb N}}$ with ${\bar d(E)>0}$ there exist ${x,n\in{\mathbb N}}$ such that (1) holds.
• The polynomials ${f_1,\dots,f_k}$ are jointly intersective, i.e., for every ${m\in{\mathbb N}}$ there exists ${n\in{\mathbb N}}$ such that ${f_1(n),\dots,f_k(n)}$ are all divisible by ${m}$.

In the same year, Frantzikinakis and Wierdl obtained the first results in the case where the ${f_i}$ are (rounded) Hardy field functions. These results were later extended by Frantzikinakis who showed that any set ${E\subset{\mathbb N}}$ with ${\bar d(E)>0}$ contains a pattern of the form

$\displaystyle \{x,x+\lfloor f_1(n)\rfloor,\dots,x+\lfloor f_k(n)\rfloor\} \ \ \ \ \ (2)$

if the ${f_i}$ are Hardy field functions of different growth, or are all multiples of a single function (plus a growth condition). As an example, Frantzikinakis result holds when each of the ${f_i}$ is a polynomial-like expression of the form ${a_1x^{c_1}+\cdots+a_kx^{c_k}}$ where ${a_1,\dots,a_k\in{\mathbb R}}$, ${c_1>\cdots>c_k}$ are positive real numbers and ${c_1\notin{\mathbb N}}$. For the functions to have different growth it suffices that the “degree” (i.e. the largest exponent, ${c_1}$) is different for each ${f_i}$.

Our new results allow for a similar conclusion (that any set ${E\subset{\mathbb N}}$ with ${\bar d(E)>0}$ contains a pattern of the form (2)) when the ${f_i}$ are arbitrary polynomial-like functions of the form ${a_1x^{c_1}+\cdots+a_k^{c_k}}$ with ${a_1,\dots,a_k,c_1,\dots,c_k\in{\mathbb R}}$, ${c_i>0}$ (and in fact our result applies in a much greater generality of Hardy field functions satisfying certain weak conditions). As a corollary we also obtain a satisfying extension of our previous results mentioned in this post.

Theorem 2 Let ${c_1,\dots,c_k}$ be positive real numbers such that no ${c_i}$ is an integer, and no difference ${c_i-c_j}$ is an integer for ${i\neq j}$. Then for every ${E\subset{\mathbb N}}$ with ${\bar d(E)>0}$, the set of ${n\in{\mathbb N}}$ for which ${E}$ contains a pattern of the form ${\{x,x+\lfloor n^{c_1}\rfloor,\dots,x+\lfloor n^{c_k}\rfloor\}}$ is thick (i.e., it contains arbitrarily long intervals).

The case ${k=1}$ follows from our earlier paper.

— 2. Strategy of the proof —

The proof follows a strategy already employed by Bergelson-Leibman-Lesigne and by Frantzikinakis and Wierdl, hinging on the Host-Kra structure theory. The first step is to use Furstenberg’s correspondence principle to transform the problem of finding patterns in sets of natural numbers with positive density into a multiple recurrence statement in ergodic theory.

The next step is to, using the structure theory, reduce the ergodic theoretic problem to a special class of measure preserving systems: nilsystems. These are compact homogeneous spaces for nilpotent Lie groups and hence have a much nicer algebraic structure than a general abstract measure preserving system. This reduction is perhaps the most novel part of the argument. Like in our previous paper we had to resort to using weighted means, instead of the more standard Cesàro means, and this introduced several new technical challenges.

Once we restrict attention to nilsystems, it is possible to obtain multiple recurrence if one has some control on the behavior of the orbits of individual points (or, more precisely, the subset of the orbit at the times prescribed by the functions ${f_i}$). These orbits are difficult to understand due to the presence of the floor function, so the idea at this point is to deduce enough information about the orbits “with floors” from a description of the orbits “without the floors”. This part is also technical, involving a mixture of known tricks and new.

Finally, the description of the orbits that we need (“without the floors”) is provided by the very recent paper of Richter. This was the main ingredient that wasn’t available until now.

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