Iterated differences sets, Diophantine approximations and applications

Abstract

Let v be an odd real polynomial (i.e. a polynomial of the form ${\sum }_{j=1}^{\ell }{a}_j{x}^{2j-1}$). We utilize sets of iterated differences to establish new results about sets of the form $\mathcal{R}(v,ϵ)=\{n\in \mathbb{N}|\Vert v(n)\Vert <ϵ\}$ where $\Vert \cdot \Vert$ denotes the distance to the closest integer. We then apply the new Diophantine results to obtain applications to ergodic theory and combinatorics. In particular, we obtain a new characterization of weakly mixing systems as well as a new variant of Furstenberg-Sárközy theorem.

Introduction

The goal of this paper is to establish new results pertaining to Diophantine inequalities involving odd real polynomials and to obtain some applications to combinatorial number theory and ergodic theory.

Assume that v is a real polynomial, with $\mathrm{deg}(v)\ge 1$, satisfying $v(0)=0$ and let $ϵ>0$. Consider the set

$$\mathcal{R}(v,ϵ)=\{n\in \mathbb{N}=\{1,2,...\}|\Vert v(n)\Vert <ϵ\},$$

where $\Vert \cdot \Vert$ denotes the distance to the nearest integer.

It is well known that sets of the form $\mathcal{R}(v,ϵ)$ are large in more than one sense. For example, it follows from Weyl's equidistribution theorem (see [23]) that $\mathcal{R}(v,ϵ)$ has positive natural density. One can also show that $\mathcal{R}(v,ϵ)$ is syndetic ([12, Theorem 1.21]), meaning that finitely many translations of $\mathcal{R}(v,ϵ)$ cover $\mathbb{N}$ (i.e. $\mathcal{R}(v,ϵ)$ has “bounded gaps”). As a matter of fact, the sets $\mathcal{R}(v,ϵ)$ posses a stronger property which is called IP^⁎. A set $E\subseteq \mathbb{N}$ is called an IP set if it contains a set of the form

$$\text{FS}({(n_k)}_{k\in \mathbb{N}})=\{n_{k_1}+\cdots +n_{k_m}|k_1<\cdots <k_m;m\in \mathbb{N}\},$$

for some increasing sequence ${(n_k)}_{k\in \mathbb{N}}$. A set $E\subseteq \mathbb{N}$ is IP^⁎ if it has a non-trivial intersection with every IP set.¹

One can show with the help of Hindman's theorem² that IP^⁎ sets have the finite intersection property, meaning that if $E_1,...,E_m\subseteq \mathbb{N}$ are IP^⁎ sets, then ${\bigcap }_{j=1}^m{E}_j$ is also IP^⁎.

When v is linear, $\mathcal{R}(v,ϵ)$ has an even stronger property than IP^⁎. Namely that of ${\mathrm{\Delta }}^{⁎}$. A set $E\subseteq \mathbb{N}$ is called a ${\mathrm{\Delta }}^{⁎}$ set if for any increasing sequence ${(n_k)}_{k\in \mathbb{N}}$, there exist $i<j$ for which

$$n_j-n_i\in E.$$

It is not hard to show that every ${\mathrm{\Delta }}^{⁎}$ set is IP^⁎. Moreover, the family of IP^⁎ sets strictly contains the family of ${\mathrm{\Delta }}^{⁎}$ sets. For example, the set

$$\mathbb{N}\setminus \{2^j-2^i|i,j\in \mathbb{N},i<j\}$$

is IP^⁎ but not ${\mathrm{\Delta }}^{⁎}$ (see [5, p. 165]).

One can show, with the help of Ramsey's Theorem, that ${\mathrm{\Delta }}^{⁎}$ sets have the finite intersection property (see [12, p. 179]). This implies, in particular, that for any ${\alpha }_1,...,{\alpha }_m\in \mathbb{R}$ and any $ϵ>0$, the set ${\bigcap }_{j=1}^m\{n\in \mathbb{N}|\Vert n{\alpha }_j\Vert <ϵ\}$ is ${\mathrm{\Delta }}^{⁎}$.

Unfortunately, for polynomials of degree two, the sets $\mathcal{R}(v,ϵ)$ are no longer ${\mathrm{\Delta }}^{⁎}$ (see, for example, [12, pp. 177-178]). One is tempted to conjecture that the ${\mathrm{\Delta }}_2^{⁎}$ sets, namely sets intersecting any set of the form

$$\{(n_{k_4}-n_{k_3})-(n_{k_2}-n_{k_1})|k_4>k_3>k_2>k_1\},$$

could be useful in dealing with polynomials of degree two and the corresponding sets $\mathcal{R}(v,ϵ)$. However, one can show, by using a natural modification of the construction in [12], that there exists $ϵ>0$ such that for any irrational α, the set $\{n\in \mathbb{N}|\Vert n^2\alpha \Vert <ϵ\}$ is not a ${\mathrm{\Delta }}_2^{⁎}$ set.

To see this, fix an irrational number α and let ${(n_k)}_{k\in \mathbb{N}}$ be an increasing sequence³ in $\mathbb{N}$ such that

$$\underset{k\to \infty }{\lim }\Vert n_k\alpha \Vert =0\text{ and }\underset{k\to \infty }{\lim }\Vert n_k^2\alpha -\frac{1}{3}\Vert =0.$$

By passing, if needed, to a subsequence, we can also assume that for any $j,k\in \mathbb{N}$ with $j<k$,

$$\Vert n_j{n}_k\alpha \Vert <\frac{1}{k}.$$

So, for any large enough and distinct $j,k\in \mathbb{N}$, we have $\Vert n_j{n}_k\alpha \Vert <\frac{ϵ}{16}$ and $\Vert n_k^2\alpha -\frac{1}{3}\Vert <\frac{ϵ}{16}$. It follows by a simple calculation that for large enough $k_4>k_3>k_2>k_1$,

$$\Vert {[(n_{k_4}-n_{k_3})-(n_{k_2}-n_{k_1})]}^2\alpha -\frac{4}{3}\Vert <ϵ,$$

which implies that the set $\mathcal{R}(n^2\alpha ,\frac{1}{6})$ is not ${\mathrm{\Delta }}_2^{⁎}$.

It comes as a pleasant surprise that ${\mathrm{\Delta }}_2^{⁎}$ sets work well with the sets $\mathcal{R}(n^3\alpha ,ϵ)$.

Proposition 1.1

For any real number α and any $ϵ>0$, the set

$$\mathcal{R}(n^3\alpha ,ϵ)=\{n\in \mathbb{N}|\Vert n^3\alpha \Vert <ϵ\}$$

is ${\mathit{\Delta }}_2^{⁎}$.

It turns out that Proposition 1.1 generalizes nicely to odd real polynomials, namely polynomials of the form

$$v(x)=\sum _{j=1}^{\ell }{a}_j{x}^{2j-1}.$$

(Note that a real polynomial v satisfies $v(-x)=-v(x)$ if and only if v is of the form (5)).

Before formulating a generalization of Proposition 1.1 to odd polynomials of arbitrary degree, we have to introduce the family of ${\mathrm{\Delta }}_{\ell }^{⁎}$ sets, $\ell \in \mathbb{N}$.

Define the function $\partial :{\bigcup }_{\ell \in \mathbb{N}}{\mathbb{Z}}^{2^{\ell }}\to \mathbb{Z}$ recursively by the formulas:

• $\partial (m_1,m_2)=m_2-m_1$.

• $\partial (m_1,...,m_{2^{\ell }})=\partial (m_{2^{\ell -1}+1},...,m_{2^{\ell }})-\partial (m_1,...,m_{2^{\ell -1}})$, $\ell >1$.

Given $\ell \in \mathbb{N}$, we will say that a set $E\subseteq \mathbb{N}$ is ${\mathrm{\Delta }}_{\ell }^{⁎}$ if for any increasing sequence ${(n_k)}_{k\in \mathbb{N}}$ in $\mathbb{N}$, there exist

$$k_1<k_2<k_3<\cdots <k_{2^{\ell }}$$

for which⁴

$$\partial (n_{k_1},...,n_{k_{2^{\ell }}})\in E.$$

For example, a set $E\subseteq \mathbb{N}$ is ${\mathrm{\Delta }}_3^{⁎}$ if for any increasing sequence ${(n_k)}_{k\in \mathbb{N}}$ in $\mathbb{N}$, there exist $k_1<\cdots <k_8$ for which

$$[(n_{k_8}-n_{k_7})-(n_{k_6}-n_{k_5})]-[(n_{k_4}-n_{k_3})-(n_{k_2}-n_{k_1})]\in E.$$

One can show that for each $\ell \in \mathbb{N}$, ${\mathrm{\Delta }}_{\ell }^{⁎}$ sets have the finite intersection property. (See Section 2 for more information on ${\mathrm{\Delta }}_{\ell }^{⁎}$ sets.)

We are now in position to state a generalization of Proposition 1.1.

Theorem 1.2

For any odd real polynomial $v(x)={\sum }_{j=1}^{\ell }{a}_j{x}^{2j-1}$ and any $ϵ>0$, the set

$$\mathcal{R}(v,ϵ)=\{n\in \mathbb{N}|\Vert v(n)\Vert <ϵ\}$$

is ${\mathit{\Delta }}_{\ell }^{⁎}$.

Remark 1.3

One can show that for $\ell >1$, the families IP^⁎ and ${\mathrm{\Delta }}_{\ell }^{⁎}$ are, so to say, in general position. Namely, IP${}^{⁎}⊈{\mathrm{\Delta }}_{\ell }^{⁎}$ (see Lemma 8.3) and ${\mathrm{\Delta }}_{\ell }^{⁎}⊈{\text{IP}}^{⁎}$ (see Theorem 8.13).

The following theorem shows that odd real polynomials are, roughly, the only polynomials for which the sets $\mathcal{R}(v,ϵ)$ are always ${\mathrm{\Delta }}_{\ell }^{⁎}$:

Theorem 1.4

Let $\ell \in \mathbb{N}$ and let $v(x)$ be a real polynomial. The set $\mathcal{R}(v,ϵ)$ is ${\mathit{\Delta }}_{\ell }^{⁎}$ for any $ϵ>0$ if and only if there exists a polynomial $w\in \mathbb{Q}[x]$ with $w(0)\in \mathbb{Z}$ and such that $v-w$ is an odd polynomial of degree at most $2\ell -1$.

There are two basic approaches to the proof of Theorem 1.2. The first approach is based on the inductive utilization of (the finite) Ramsey Theorem. The second approach uses a special family of ultrafilters in $\beta \mathbb{N}$ which is of interest in its own right and has not been utilized before in a similar context. Each of these approaches has its own pros and cons.

The first approach allows to formulate and prove a finitistic version of Theorem 1.2 (this is a pro), but the proof gets quite cumbersome (this is a con). This approach is carried out in Subsection 3.2.

The second approach, which is implemented in Subsection 3.1, has the advantage of being shorter and much easier to follow. The disadvantage of this approach seems to be mostly lying with the fact that some readers may not be familiar with ultrafilters. We remedy this by giving detailed definitions and some of the necessary background in Section 2.

It is worth mentioning that we will also utilize the ultrafilter technique in the proofs of Theorem 1.4 (see Section 5) and of a converse to Theorem 1.2 (see Section 4).

In Section 6, we deal with applications to unitary actions. In particular, we establish the following result.

Theorem 1.5

Let $U:\mathcal{H}\to \mathcal{H}$ be a unitary operator and let $v(x)={\sum }_{j=1}^{\ell }{a}_j{x}^{2j-1}$ be a non-zero odd polynomial with $v(\mathbb{Z})\subseteq \mathbb{Z}$. The following are equivalent:

• U has discrete spectrum (i.e. $\mathcal{H}$ is spanned by eigenvectors of U).

• For any $f\in \mathcal{H}$ and any $ϵ>0$, the set

  $$\{n\in \mathbb{N}|{\Vert U^{v(n)}f-f\Vert }_{\mathcal{H}}<ϵ\}$$

  is ${\mathit{\Delta }}_{\ell }^{⁎}$.

Theorem 1.5 has the following ergodic-theoretical corollary.

Corollary 1.6

Let $(X,\mathcal{A},\mu )$ be a probability space⁵ and let $T:X\to X$ be an ergodic invertible probability measure preserving transformation. The following are equivalent:

• $(X,\mathcal{A},\mu ,T)$ is isomorphic to a translation on a compact abelian group.

• For any odd polynomial $v(x)={\sum }_{j=1}^{\ell }{a}_j{x}^{2j-1}$ with $v(\mathbb{Z})\subseteq \mathbb{Z}$, any $A\in \mathcal{A}$ and any $ϵ>0$, the set

  $$\{n\in \mathbb{N}|\mu (A\cap T^{-v(n)}A)>\mu (A)-ϵ\}$$

  is ${\mathit{\Delta }}_{\ell }^{⁎}$.

• There exists a non-zero odd polynomial $v(x)={\sum }_{j=1}^{\ell }{a}_j{x}^{2j-1}$ with $v(\mathbb{Z})\subseteq \mathbb{Z}$ such that for any $A\in \mathcal{A}$ and any $ϵ>0$, the set

  $$\{n\in \mathbb{N}|\mu (A\cap T^{-v(n)}A)>\mu (A)-ϵ\}$$

  is ${\mathit{\Delta }}_{\ell }^{⁎}$.

Another application of Theorem 1.5 to measure preserving systems requires the introduction of the notion of an almost ${\mathrm{\Delta }}_{\ell }^{⁎}$ set, denoted by A-${\mathrm{\Delta }}_{\ell }^{⁎}$. Recall that the upper Banach density of a set $E\subseteq \mathbb{N}$, $d^{⁎}(E)$, is defined by

$$d^{⁎}(E)=\underset{N-M\to \infty }{\lim \sup }\frac{|E\cap \{M+1,...,N\}|}{N-M},$$

where, for a finite $F\subseteq \mathbb{N}$, $|F|$ denotes the cardinality of F. Given $\ell \in \mathbb{N}$, a set $D\subseteq \mathbb{N}$ is A-${\mathrm{\Delta }}_{\ell }^{⁎}$ if there exists a set $E\subseteq \mathbb{N}$ with $d^{⁎}(E)=0$, such that $D\cup E$ is ${\mathrm{\Delta }}_{\ell }^{⁎}$.

Theorem 1.7

Let $(X,\mathcal{A},\mu ,T)$ be an invertible probability measure preserving system and let $v(x)={\sum }_{j=1}^{\ell }{a}_j{x}^{2j-1}$ be an odd polynomial with $v(\mathbb{Z})\subseteq \mathbb{Z}$. For any $A\in \mathcal{A}$ and any $ϵ>0$, the set

$${\mathcal{R}}_A(v,ϵ)=\{n\in \mathbb{N}|\mu (A\cap T^{-v(n)}A)>{\mu }^2(A)-ϵ\}$$

is A-${\mathit{\Delta }}_{\ell }^{⁎}$.

Remark 1.8

It was shown in [6] that the “sets of large returns” ${\mathcal{R}}_A(v,ϵ)$ have the IP^⁎ property for any polynomial v with $v(\mathbb{Z})\subseteq \mathbb{Z}$ and satisfying $v(0)=0$. It will be shown in Section 8 that for each $\ell \in \mathbb{N}$, there exists an IP^⁎ set which is not A-${\mathrm{\Delta }}_{\ell }^{⁎}$. So, Theorem 1.7 provides new information about sets of large returns when v is an odd polynomial.

We remark that the quantity ${\mu }^2(A)$ in (6) is optimal (consider any strongly mixing system⁶).

The following corollary of Theorem 1.7 is a result in additive combinatorics which might be seen as a variant of the Furstenberg-Sárközy theorem (see [20] and [12, Theorem 3.16]).

Corollary 1.9

Let $E\subseteq \mathbb{N}$ be such that $d^{⁎}(E)>0$ and let $v(x)={\sum }_{j=1}^{\ell }{a}_j{x}^{2j-1}$ be an odd polynomial with $v(\mathbb{Z})\subseteq \mathbb{Z}$. Then the set

$$\{n\in \mathbb{N}|v(n)\in E-E\}$$

is A-${\mathit{\Delta }}_{\ell }^{⁎}$.

We also have a new recurrence property for weakly mixing systems. A probability measure preserving system $(X,\mathcal{A},\mu ,T)$ is weakly mixing if for any $A,B\in \mathcal{A}$,

$$\underset{N\to \infty }{\lim }\frac{1}{N}\sum _{j=1}^N|\mu (A\cap T^{-n}B)-\mu (A)\mu (B)|=0.$$

Corollary 1.10

Let $v(x)={\sum }_{j=1}^{\ell }{a}_j{x}^{2j-1}$ be a non-zero odd polynomial with $v(\mathbb{Z})\subseteq \mathbb{Z}$. An invertible probability measure preserving system $(X,\mathcal{A},\mu ,T)$ is weakly mixing if and only if for any $A,B\in \mathcal{A}$ and any $ϵ>0$, the set

$${\mathcal{R}}_{A,B}(v,ϵ)=\{n\in \mathbb{N}||\mu (A\cap T^{-v(n)}B)-\mu (A)\mu (B)|<ϵ\}$$

is A-${\mathit{\Delta }}_{\ell }^{⁎}$.

In Section 7, we provide an example of a weakly mixing system $(X,\mathcal{A},\mu ,T)$ which shows that in the statement of Corollary 1.10, A-${\mathrm{\Delta }}_{\ell }^{⁎}$ can not be replaced by ${\mathrm{\Delta }}_{\ell }^{⁎}$.

We conclude the introduction with formulating a recent result [10] which demonstrates yet another connection between ${\mathrm{\Delta }}_{\ell }^{⁎}$ sets and ergodic theory.

Theorem 1.11 Cf. [18]

Let $(X,\mathcal{A},\mu ,T)$ be an invertible probability measure preserving system. The following are equivalent:

• $(X,\mathcal{A},\mu ,T)$ is strongly mixing.

• There exists an $\ell \in \mathbb{N}$ such that for any $A\in \mathcal{A}$ and any $ϵ>0$, the set

  $$\{n\in \mathbb{N}||\mu (A\cap T^{-n}A)-{\mu }^2(A)|<ϵ\}$$

  is ${\mathit{\Delta }}_{\ell }^{⁎}$.

• For any $\ell \in \mathbb{N}$, any $A_0,...,A_{\ell +1}\in \mathcal{A}$ and any $ϵ>0$, the set

  $$\{n\in \mathbb{N}||\mu (A_0\cap T^{-n}{A}_1\cdots \cap T^{-(\ell +1)n}{A}_{\ell +1})-\prod _{j=0}^{\ell +1}\mu (A_j)|<ϵ\}$$

  is ${\mathit{\Delta }}_{\ell }^{⁎}$.

The structure of the paper is as follows. In Section 2, we provide the necessary background on ultrafilters and establish the connection between ultrafilters and ${\mathrm{\Delta }}_{\ell }^{⁎}$ sets. In Section 3, we prove Theorem 1.2 as well as its finitistic version. In Section 4, we prove a converse to Theorem 1.2. In Section 5, we prove Theorem 1.4. In Section 6, we focus on applications to unitary actions. In Section 7, we provide an example of a weakly mixing system which demonstrates that Corollary 1.10 can not be improved. In Section 8, we discuss the relations between the various families of subsets of $\mathbb{N}$ that we deal with throughout this paper.