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Infinite sumsets in $U^k(Φ)$-uniform sets

Tristán Radić

TL;DR

The paper extends infinite sumset patterns from positive-density sets to $U^k(Φ)$-uniform subsets of ${\mathbb N}$ by leveraging Host–Kra local uniformity seminorms and the pronilfactor structure of ergodic systems. It develops a dynamic framework via Furstenberg correspondence and open pronilfactor extensions to translate uniformity into return-time configurations, proving a main dynamical theorem that yields rich sumset configurations along parallelepiped vertices and higher-order parity obstructions. By intersecting with Nil-Bohr$_0$ sets and analyzing joinings of pronilfactors, the work derives stronger combinatorial consequences and links to regionally proximal relations $\operatorname{RP}^{[k]}(X)$ in topological dynamics. The results unify techniques from ergodic theory, nilsystems, and additive combinatorics to characterize when infinite sumset patterns can be embedded in $U^k(Φ)$-uniform sets, with implications for parity obstructions and disintegration-driven analyses.

Abstract

Extending recent developments of Kra, Moreira, Richter and Roberson, we study infinite sumset patterns in $U^k(Φ)$-uniform subsets of the integers, defined via the local uniformity seminorms introduced by Host and Kra. The main result relates the degree $k$ of a $U^k(Φ)$-uniform set to the existence of sumset patterns along prescribed vertices of $\ell$-dimensional parallelepipeds, for $k \leq \ell$. The proof relies on a dynamical analysis of return-time sets to neighborhoods of points lying over pronilfactor fibers. We then derive higher-order parity obstructions for sumset patterns and consequences in topological dynamics.

Infinite sumsets in $U^k(Φ)$-uniform sets

TL;DR

The paper extends infinite sumset patterns from positive-density sets to -uniform subsets of by leveraging Host–Kra local uniformity seminorms and the pronilfactor structure of ergodic systems. It develops a dynamic framework via Furstenberg correspondence and open pronilfactor extensions to translate uniformity into return-time configurations, proving a main dynamical theorem that yields rich sumset configurations along parallelepiped vertices and higher-order parity obstructions. By intersecting with Nil-Bohr sets and analyzing joinings of pronilfactors, the work derives stronger combinatorial consequences and links to regionally proximal relations in topological dynamics. The results unify techniques from ergodic theory, nilsystems, and additive combinatorics to characterize when infinite sumset patterns can be embedded in -uniform sets, with implications for parity obstructions and disintegration-driven analyses.

Abstract

Extending recent developments of Kra, Moreira, Richter and Roberson, we study infinite sumset patterns in -uniform subsets of the integers, defined via the local uniformity seminorms introduced by Host and Kra. The main result relates the degree of a -uniform set to the existence of sumset patterns along prescribed vertices of -dimensional parallelepipeds, for . The proof relies on a dynamical analysis of return-time sets to neighborhoods of points lying over pronilfactor fibers. We then derive higher-order parity obstructions for sumset patterns and consequences in topological dynamics.
Paper Structure (27 sections, 42 theorems, 80 equations)

This paper contains 27 sections, 42 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Dynamical Perspective
  3. Notation
  4. Preliminaries
  5. Dynamical systems
  6. Pronilfactors and Host-Kra seminorms
  7. Local uniformity seminorms
  8. Cubic spaces
  9. Bounded marginals and lift of measures
  10. Reduction to Dynamics
  11. Furstenberg correspondence principle
  12. Extension with open topological pronilfactor
  13. Dynamic reformulation of Theorems \ref{['theorem uniformity BCD']} and \ref{['theorem uniformity BB']}
  14. Erdős cubes and Erdős progressions
  15. Main dynamical result
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $k \geq 2$ be an integer and $A_1, \ldots, A_k \subset {\mathbb N}$ be $U^k(\Phi)$-uniform sets with respect to the same Fø lner sequence $\Phi$. Then for all integers $1 \leq \ell_1 < \ell_2 < \cdots < \ell_k \leq \ell$, there exist infinite sets $B_1 , \ldots, B_\ell \subset {\mathbb N}$ such

Theorems & Definitions (86)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1
  • Lemma 1
  • proof
  • Proposition 1
  • Remark 1
  • Corollary 1
  • proof
  • ...and 76 more