On the interplay between additive and multiplicative largeness and its combinatorial applications

Abstract

Many natural notions of additive and multiplicative largeness arise from results in Ramsey theory. In this paper, we explain the relationships between these notions for subsets of $\mathbb{N}$ and in more general ring-theoretic structures. We show that multiplicative largeness begets additive largeness in three ways and give a collection of examples demonstrating the optimality of these results. We also give a variety of applications arising from the connection between additive and multiplicative largeness. For example, we show that given any $n,k\in \mathbb{N}$, any finite set with fewer than n elements in a sufficiently large finite field can be translated so that each of its elements becomes a non-zero kth power. We also prove a theorem concerning Diophantine approximation along multiplicatively syndetic subsets of $\mathbb{N}$ and a theorem showing that subsets of positive upper Banach density in certain multiplicative sub-semigroups of $\mathbb{N}$ of zero density contain arbitrarily long arithmetic progressions. Along the way, we develop a new characterization of upper Banach density in a wide class of amenable semigroups and make explicit the uniformity in recurrence theorems from measure theoretic and topological dynamics. This in turn leads to strengthened forms of classical theorems of Szemerédi and van der Waerden on arithmetic progressions.

Introduction

A classic result of van der Waerden [56] states that at least one cell of any finite partition of $\mathbb{N}=\{1,2,3,\dots \}$ contains arbitrarily long arithmetic progressions. A far-reaching generalization by Rado [52] characterizes those systems of homogeneous linear equations to which at least one cell of any finite partition of $\mathbb{N}$ contains a solution. Underlying both of these foundational results in Ramsey theory are notions of “additive largeness,” and for many of these notions of largeness there are two essential theorems: at least one cell of any partition of $\mathbb{N}$ is additively large, and additively large sets contain the sought-after combinatorial configurations.

The pertinent notions of largeness underlying van der Waerden's theorem are syndeticity, piecewise syndeticity, and additive upper Banach density. In Rado's theorem, the notions of IP structure and centrality play a fundamental role. We will be concerned with these notions in this work.¹ To facilitate the discussion in the introduction, we define them now for subsets $A\subseteq \mathbb{N}$.

• A is syndetic if there exists a finite set $F\subseteq \mathbb{N}$ for which

  $$\bigcup _{f\in F}(A-f)=\mathbb{N},\text{where }A-f:=\{n\in \mathbb{N}|n+f\in A\}.$$

• A is piecewise syndetic if there exists a syndetic set $S\subseteq \mathbb{N}$ and a sequence ${(m_n)}_{n\in \mathbb{N}}\subseteq \mathbb{N}$ for which

  $$S\cap \bigcup _{n=1}^{\infty }\{m_n+1,\dots ,m_n+n\}\subseteq A.$$

• A has positive additive upper Banach density if

  $$\begin{array}{cc}\hfill \underset{N-M\to \infty }{\lim \sup }\frac{|A\cap \{N,\dots ,N+M-1\}|}{N-M}& :=\hfill \\ \hfill \underset{n\to \infty }{\lim \sup }\underset{m\in \mathbb{N}}{\max }\frac{|A\cap (m+\{1,\dots ,n\})|}{n}& >0.\hfill \end{array}$$

• A is AP-rich if it contains arbitrarily long arithmetic progressions $\{x,x+y,x+2y,\dots ,x+ky\}$.

• A is an IP set if it contains a set of the form

  $$\text{FS}{(n_i)}_{i\in \mathbb{N}}=\left\{\sum _{i\in I}{n}_i\right|\text{finite, non-empty }I\subseteq \mathbb{N}\},{(n_i)}_{i\in \mathbb{N}}\subseteq \mathbb{N}.$$

• A is central² if it is a member of an additively minimal idempotent ultrafilter in $(\beta \mathbb{N},+)$.³

Many fundamental results in Ramsey theory can be interpreted as elucidating the relationships between these notions of largeness. For example, that syndetic sets are AP-rich is a consequence of van der Waerden's theorem. Hindman's theorem [38] implies that if A piecewise syndetic, then $A-n$ is an IP set for some $n\in \mathbb{N}$, and it follows by a remark of Furstenberg [32, Page 163] that $A-n$ is central for some $n\in \mathbb{N}$.⁴ Szemerédi's theorem [54] gives that sets of positive upper Banach density are AP-rich, and Furstenberg and Katznelson's IP Szemerédi theorem [28] (Theorem 7.5 below) gives that in any set A of positive upper Banach density, for every $\ell \in \mathbb{N}$, the set of step sizes of arithmetic progressions of length ℓ appearing in A has non-empty intersection with every IP set.

The set of positive integers supports another associative and commutative operation, multiplication, which is just as fundamental to its structure. For each of the notions of additive largeness defined above, there is a natural analogous notion of multiplicative largeness for $A\subseteq \mathbb{N}$. Fix $p_1,p_2,\dots$ an enumeration of the prime numbers.

• A is multiplicatively syndetic if there exists a finite set $F\subseteq \mathbb{N}$ for which

  $$\bigcup _{f\in F}(A/f)=\mathbb{N},\text{where }A/f:=\{n\in \mathbb{N}|nf\in A\}.$$

• A is multiplicatively piecewise syndetic if there exists a multiplicatively syndetic set $S\subseteq \mathbb{N}$ and a sequence ${(m_n)}_{n\in \mathbb{N}}\subseteq \mathbb{N}$ for which

  $$S\cap \bigcup _{n=1}^{\infty }\{m_n{p}_1^{e_1}\cdots p_n^{e_n}|e_1,\dots ,e_n\in \{1,\dots ,n\}\}\subseteq A.$$

• A has positive multiplicative upper Banach density⁵ if

  $$\underset{n\to \infty }{\lim \sup }\underset{m\in \mathbb{N}}{\max }\frac{|A\cap \{m{p}_1^{e_1}\cdots p_n^{e_n}|e_1,\dots ,e_n\in \{1,\dots ,n\}\}|}{n^n}>0.$$

• A is GP-rich if it contains arbitrarily long geometric progressions $\{x,xy,x{y}^2,\dots ,x{y}^k\}$.

• A is a multiplicative IP set if it contains a set of the form

  $$\text{FP}{(n_i)}_{i\in \mathbb{N}}=\left\{\prod _{i\in I}{n}_i\right|\text{finite, non-empty }I\subseteq \mathbb{N}\},{(n_i)}_{i\in \mathbb{N}}\subseteq \mathbb{N}.$$

• A is multiplicatively central if it is a member of a multiplicatively minimal idempotent ultrafilter in $(\beta \mathbb{N},\cdot )$.

Multiplicative analogues of the Ramsey theoretical results described above explain some of the relationships between these notions of multiplicative largeness. For example, a multiplicative analogue of Hindman's theorem [12, Lemma 2.1] gives that if A multiplicatively piecewise syndetic, then $A/n$ is a multiplicative IP set for some $n\in \mathbb{N}$, and an analogue of Szemerédi's theorem gives that sets of positive multiplicative upper Banach density are GP-rich (cf. [7, Theorem 3.8]).

With addition and multiplication occupying a central role in the study of the positive integers, it is incumbent on us to explain the relationships between these notions of additive and multiplicative largeness. It has been known for some time that additive and multiplicative largeness can be simultaneously guaranteed in one cell of any partition of $\mathbb{N}$; this was first explained by Hindman [39] and later developed further in [12]. Such partition results leave open the question as to how these notions of additive and multiplicative largeness interact: does one beget the other, or are the notions in general position and hence guaranteed to co-exist with potentially no relation?

A natural first question exploring the direct relationship between additive and multiplicative largeness is: to what extent can a set of positive integers be additively large but multiplicatively small? The set $(4\mathbb{N}-2)\cup \text{FS}{\left(2^{2^i}\right)}_{i\in \mathbb{N}}$ is additively large (both syndetic and an IP set) but multiplicatively small (neither of positive multiplicative density nor a multiplicative IP set). In searching for a complementary example – that is, a set which is multiplicatively large but additively small – one encounters an asymmetry between the relationships amongst notions of additive and multiplicative largeness: multiplicatively large sets cannot be additively very small. Several instances of this principle have appeared in the literature, and we cite three of them here.⁶

Theorem A [9, Lemma 5.11]

For all $A\subseteq \mathbb{N}$, if A is multiplicatively syndetic, then A is additively central.

Theorem B [13, Theorem 3.5], [23, Proposition 4.1]

For all $A\subseteq \mathbb{N}$, if A is multiplicatively central, then A is additively ${\mathit{\text{IP}}}_{\mathit{\text{0}}}$.⁷

Theorem C [7, Theorems 3.2 & 3.15]

For all $A\subseteq \mathbb{N}$, if A has positive multiplicative upper Banach density, then A is AP-rich; what is more, A contains arbitrarily large “geo-arithmetic” patterns.⁸

Thus, while an additively syndetic IP set need not have positive multiplicative density nor be a multiplicative IP set, it follows from Theorem A that a multiplicatively syndetic set necessarily has positive additive upper Banach density and is an additive IP set.

In spite of these results, the relationships between notions of additive and multiplicative largeness are not entirely understood. Attempting to better understand these relationships quickly leads to interesting open problems. For example, it is unknown whether or not additively syndetic subsets of $\mathbb{N}$ are necessarily GP-rich. While we only discuss this particular problem in passing (see Section 9.1), we advance our understanding of these relationships in this paper and obtain new and interesting applications in the theory of amenability and invariant means on semigroups, Diophantine approximation, and combinatorial number theory.

In this paper, we wish to explore in the proper generality the interplay between notions of additive and multiplicative largeness. The natural numbers $\mathbb{N}$ with addition and multiplication form a semiring: a set S supporting a commutative additive semigroup $(S,+)$ and a multiplicative semigroup $(S,\cdot )$ in which distributivity holds: $s(s_1+s_2)=s{s}_1+s{s}_2$, and similarly on the right. Semirings are minimally structured algebraic objects in which addition and multiplication co-exist, and it is in them that we study the relationships between additive and multiplicative largeness.

This work is comprised of two parts. First, we generalize Theorem A, Theorem B, Theorem C to the context of semirings, in the process strengthening Theorem B for subsets of $\mathbb{N}$. Thus, we prove the following theorems for semirings $(S,+,\cdot )$ that contain a suitable multiplicative subsemigroup $(R,\cdot )$; the requisite definitions can be found in Sections 2, 4, and 7, and the meaning of “suitable” can be found in the precise statements of the theorems in the sections to which they belong.

Theorem 6.1

For all $A\subseteq R$, if A is multiplicatively syndetic, then A is additively central.

Theorem 7.1

For all $A\subseteq R$, if A is multiplicatively piecewise syndetic, then A is additively ${\mathit{\text{IP}}}_{\mathit{\text{0}}}$.⁹

Theorems 8.2 and 8.8

For all $A\subseteq R$, if A has positive multiplicative upper Banach density, then A is additively combinatorially rich; what is more, A contains arbitrarily large “geo-arithmetic” patterns.

In Section 9, we provide extremal examples that serve to illustrate the optimality of these results.

Useful throughout the paper is a new characterization of upper Banach density, developed in Section 3, that is applicable in a wide class of amenable semigroups. The upper Banach densities in (1.1) and (1.2) are defined with the help of Følner sequences (see Definition 3.4); in a general left amenable semigroup $(S,\cdot )$, we define the upper Banach density of $A\subseteq S$ to be

$$d^{⁎}(A)=\sup \{\lambda ({\mathbb{1}}_A)|\lambda \text{ a left translation invariant mean on }(S,\cdot )\},$$

(see Definition 2.7) and we prove the following result.

Theorem 3.5

Let $(S,\cdot )$ be an amenable group or a commutative semigroup. For all $A\subseteq S$,

$$d^{⁎}(A)=\sup \{\delta \ge 0|\forall \mathit{\text{finite}}F\subseteq S,\exists s\in S,|F\cap A\cdot s^{-1}|\ge \delta |F|\},$$

where $A\cdot s^{-1}=\{t\in S|t\cdot s\in A\}$.

This characterization of upper Banach density not only serves as an important tool throughout the paper, it allows us to improve on recent results regarding densities defined along Følner sequences obtained in [43, Section 2].

The main impetus for this work arose not from results in the abstract semiring framework but from the diverse array of related results and applications afforded to us by the ideas behind these theorems. We proceed now with a sampling of these related results and representative applications of the theorems above. The focus in each of the following results is on the interplay between addition and multiplication.

In Section 6, we combine Theorem 6.1 and an ${\text{IP}}^{⁎}$ version of Szemerédi's theorem to prove that cosets of finite index multiplicative subgroups in infinite division rings are additively thick (cf. [8, Remark 5.23]). The following finitary analogue of that result is also achieved using the ideas behind the proof of Theorem 6.1.

Theorem 6.6

For all $n,k\in \mathbb{N}$, there exists an $N=N(n,k)\in \mathbb{N}$ with the following property. For all finite fields $\mathbb{F}$ with $|\mathbb{F}|\ge N$ and all $F\subseteq \mathbb{F}$ with $|F|\le n$, there exists a non-zero $x\in \mathbb{F}$ for which every element of $x^k+F$ is a non-zero kth power.

As another application of Theorem 6.1, we exhibit a family of additively large sets which are defined with the help of familiar multiplicative functions from number theory. Let ${\nu }_2$, $\mathrm{\Omega }:\mathbb{N}\to \mathbb{N}\cup \{0\}$ denote the 2-adic valuation and the prime divisor counting function, defined by ${\nu }_2(2^{e_1}{p}_2^{e_2}\cdots p_k^{e_k})=e_1$ and $\mathrm{\Omega }(p_1^{e_1}\cdots p_k^{e_k})={\sum }_{i=1}^k{e}_i$, where the $p_i$'s are distinct primes. Denote by $\{x\}$ the fractional part of $x\in \mathbb{R}$.

Corollary 6.13

Let $p\in \mathbb{R}[x]$ be a non-constant polynomial with irrational leading coefficient, and let $I\subseteq [0,1)$ be an interval. The sets

$$\{n\in \mathbb{N}|\{p({\nu }_2(n))\}\in I\left\}\mathit{\text{and}}\right\{n\in \mathbb{N}|\{p(\mathit{\Omega }(n))\}\in I\}$$

are multiplicatively syndetic in $(\mathbb{N},\cdot )$, hence additively central in $(\mathbb{N},+)$.

This result complements known results about the uniform distribution of sequences of the form ${(\{g(n)\})}_{n\in \mathbb{N}}$ and ${(\{p(g(n))\})}_{n\in \mathbb{N}}$ where p is a polynomial and $g:\mathbb{N}\to \mathbb{N}$ is an arithmetic function arising in number theory; see, for example, [26], [27]. Since additively central sets contain solutions to partition regular systems of homogeneous linear equations (see [32, Chapter 8] or [45, Chapter 14]), another consequence of Corollary 6.13 is that the sets in (1.3) contain solutions to such systems. Corollary 7.10 exhibits yet another application.

In Section 7, we explore the main applications of a corollary of Theorem 7.1: additive ${\mathit{\text{IP}}}_{\mathit{\text{0}}}^{⁎}$ sets – those sets which have non-empty intersection with all ${\mathit{\text{IP}}}_{\mathit{\text{0}}}$ sets – have non-empty intersection with all multiplicatively piecewise syndetic sets. Additive ${\text{IP}}_{\text{0}}^{⁎}$ sets arise naturally in recurrence theorems in measure theoretic and topological dynamics to describe the set of “return times” of sets of positive measure and open sets, and thus Theorem 7.1 enhances theorems in which these classes appear.

We demonstrate in Section 7 how to derive ${\text{IP}}_{\text{0}}^{⁎}$ versions of two prominent examples of such recurrence theorems: Furstenberg and Katznelson's IP Szemerédi theorem [28] and the polynomial IP van der Waerden theorem [16]. Then, we find a number of applications making use of the enhanced statements of these results: Theorem 6.6 and Corollary 8.10 are corollaries of the ${\text{IP}}_0^{⁎}$ version of the IP Szemerédi theorem, and the following corollary in Diophantine approximation follows from the ${\text{IP}}_0^{⁎}$ version of the polynomial IP van der Waerden theorem. Denote by $\Vert x\Vert$ the distance from x to the nearest integer.

Corollary 7.10

Let $p\in \mathbb{R}[x]$ be a non-constant polynomial with irrational leading coefficient, and let $I\subseteq [0,1)$ be an interval. For all $d\in \mathbb{N}$,

$$\underset{\begin{array}{c}\hfill 1\le n\le N\hfill \\ \hfill \{p(\mathit{\Omega }(n))\}\in I\hfill \end{array}}{\min }\Vert f(n)\Vert ⟶0\mathit{\text{as}}N\to \infty$$

uniformly in polynomials $f\in \mathbb{R}[x]$ of degree less than or equal to d with no constant term.

In Section 8, we prove that subsets of a semiring that have positive multiplicative density in an additively combinatorially rich multiplicative sub-semigroup are additively combinatorially rich. This conclusion is particularly novel in the case that the multiplicative sub-semigroup has zero additive upper Banach density since this rules out the case that the additive combinatorial richness of the subset arises from it having positive additive upper Banach density. As an example of an application, we use the theorem of Green and Tao [37] to prove in Theorem 8.7 that any subset of positive multiplicative density in the multiplicative sub-semigroup of positive integers that appear as norms of algebraic integers in a number field contains arbitrarily long arithmetic progressions. The following is a special case of this result that makes use of the ring of integers in $\mathbb{Q}(\sqrt[3]{2})$; note that the multiplicative sub-semigroup R has zero additive upper Banach density [50, Theorem T].

Theorem 8.7

Let

$$R=\{|x_1^3+2x_2^3+4x_3^3-6x_1{x}_2{x}_3||x_1,x_2,x_3\in \mathbb{Z}\}\setminus \{0\}.$$

The set $(R,\cdot )$ is a sub-semigroup of $(\mathbb{N},\cdot )$, and any set $A\subseteq R$ satisfying $d_{(R,\cdot )}^{⁎}(A)>0$ contains arbitrarily long arithmetic progressions.

We also show that subsets of positive multiplicative density in positive additive density sub-semigroups of a semigroup contain geo-arithmetic patterns. The following is an example application which, in the $k=1$ case and considering additive upper Banach density in place of multiplicative upper Banach density, is a consequence of Szemerédi's theorem in finite characteristic, [28, Theorem 9.10].

Corollary 8.10

Let $p\in \mathbb{N}$ be prime, and suppose $A\subseteq {\mathbb{F}}_p[x]\setminus \{0\}$ has positive multiplicative upper Banach density in $({\mathbb{F}}_p[x]\setminus \{0\},\cdot )$. For all $d,n\in \mathbb{N}$, there exist $y,z\in {\mathbb{F}}_p[x]$, $z\ne 0$, and a d-dimensional vector subspace V of ${\mathbb{F}}_p[x]$ such that

$$\bigcup _{v\in V}z\{y+v,{(y+v)}^2,\cdots ,{(y+v)}^k\}\subseteq A.$$

This paper is organized as follows. In Section 2 we define several notions of largeness for subsets of semigroups and discuss the hierarchy between them. The alternate characterization of upper Banach density is proved in Section 3. In Section 4 we define semirings and describe how notions of largeness behave under semigroup homomorphisms. Section 5 is a preamble to Sections 6, 7, and 8, where we prove Theorem 6.1, Theorem 7.1, Theorem 8.2, respectively, and prove the related results and applications discussed above. Finally, in Section 9, we present the extremal examples that serve to show the optimality of the main theorems. Section 10 is an index of the most frequently used symbols in the work.

Acknowledgements

The authors would like to thank Jim Cogdell and Warren Sinnott for their assistance with the material in Section 8.1 and Tim Browning for calling our attention to the work of R.W.K. Odoni. Thanks also goes to Neil Hindman for improving Lemma 4.6 and Corollary 4.7 and to Joel Moreira and Florian Richter for numerous helpful conversations and a close reading of the paper. Last but not least, the authors give thanks to the referees for numerous helpful corrections and suggestions.