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On infinite sumsets and sets of multiple recurrence

Luke Hetzel

Abstract

We answer two questions of Kra, Moreira, Richter and Robertson regarding the existence of infinite sumsets of the form $B + C$ in dense and sparse sets of integers and the relation of sumsets to sets of recurrence. We then further generalize these results, yielding new characterizations of sets of multiple measurable and topological recurrence.

On infinite sumsets and sets of multiple recurrence

Abstract

We answer two questions of Kra, Moreira, Richter and Robertson regarding the existence of infinite sumsets of the form in dense and sparse sets of integers and the relation of sumsets to sets of recurrence. We then further generalize these results, yielding new characterizations of sets of multiple measurable and topological recurrence.
Paper Structure (8 sections, 9 theorems, 39 equations)

This paper contains 8 sections, 9 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Historical Overview
  3. Sumsets and Sets of Measurable Recurrence
  4. Sumsets and Sets of Topological Recurrence
  5. Sumsets in Sets of Zero Banach Density
  6. Sets of Measurable Recurrence
  7. Sets of Topological Recurrence
  8. Sets of Zero Banach Density

Key Result

Theorem 1.1

Kra-Moreira-Richter-Robertson-sumsets-survey Let $S$ be a set of strong recurrence. Then for every $A \subset \mathbb{N}$ with $d^{*}(A) > 0$ there exist infinite sets $B \subset S, C \subset A$ with

Theorems & Definitions (18)

  • Theorem 1.1
  • Definition 1.3
  • Theorem 1
  • Corollary 1.4
  • Definition 1.5
  • Theorem 2
  • Theorem 3
  • Corollary 1.8
  • Lemma 2.1
  • proof
  • ...and 8 more