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

Vitaly Bergelson, Daniel Glasscock

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 $k^{\text{th}}$ 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.

On the interplay between notions of 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 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 , any finite set with fewer than elements in a sufficiently large finite field can be translated so that each of its elements becomes a non-zero power. We also prove a theorem concerning Diophantine approximation along multiplicatively syndetic subsets of and a theorem showing that subsets of positive upper Banach density in certain multiplicative sub-semigroups of 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.

Paper Structure

This paper contains 24 sections, 53 theorems, 116 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Results
  4. Acknowledgements
  5. Notions of largeness in semigroups
  6. Combinatorially rich sets and Hales-Jewett-type theorems
  7. An alternate characterization of upper Banach density
  8. Semirings: context for the interaction of additive and multiplicative largeness
  9. Preamble to Sections \ref{['sec:posexamplea']} through \ref{['sec:counterexamples']}
  10. Multiplicative syndeticity implies additive centrality
  11. Multiplicatively syndetic subgroups of fields
  12. Lower density and a class of syndetic sets in (N,.)
  13. Multiplicative piecewise syndeticity implies additive IP0
  14. Additive IP0* sets arising in dynamics
  15. Multiplicative density implies additive combinatorial richness
  16. ...and 9 more sections

Key Result

Lemma 2.9

Let $(S,\cdot)$ be a semigroup and $r \in \mathbb{N}$. Figure fig:containmentdiagram illustrates containment amongst classes of largeness in $(S,\cdot)$ ($\mathcal{X} \to \mathcal{Y}$ indicates that $\mathcal{X} \subseteq \mathcal{Y}$), with the position of the classes $\mathcal{D}$ and $\mathcal{D}

Figures (2)

  • Figure 1: Containment amongst classes of largeness in a semigroup.
  • Figure 2: Containment amongst additive and multiplicative classes of largeness in a semiring $(S,+,\cdot)$. A solid line $\mathcal{X}$ --- $\mathcal{Y}$ with $\mathcal{X}$ positioned above $\mathcal{Y}$ indicates that $\mathcal{X}(S,+) \subseteq \mathcal{Y}(S,+)$ and $\mathcal{X}(S,\cdot) \subseteq \mathcal{Y}(S,\cdot)$. A solid arrow $\mathcal{X} \to \mathcal{Y}$ indicates that $\mathcal{X}(R,\cdot) \subseteq \mathcal{Y}(S,+)$ for suitable subsemigroups $(R,\cdot)$ of $(S,\cdot)$. A dashed arrow $\mathcal{X} \dashrightarrow \mathcal{Y}$ indicates that $\mathcal{X}(\mathbb{N},\cdot) \not\subseteq \mathcal{Y}(\mathbb{N},+)$. See Definition \ref{['def:combrichuptoe']} for the definition of the class $\mathcal{CR}_\mathcal{E}$.

Theorems & Definitions (110)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 100 more