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

TL;DR

This paper investigates the structure of infinite sumset patterns in $U^k(\

Contribution

It introduces new connections between $U^k(\" and sumset patterns, and provides examples from well-known sequences like Thue-Morse and Rudin-Shapiro.

Findings

Relates the degree $k$ of $U^k(\

Higher order parity obstructions are established for sumsets from nilsystems.

Provides explicit examples of $U^k(\

Abstract

Extending recent developments of Kra, Moreira, Richter and Roberson, we study infinite sumset patterns in $U^k(\Phi)$-uniform subsets of the integers, defined via the local uniformity seminorms introduced by Host and Kra. We relate the degree $k$ of a $U^k(\Phi)$-uniform set to the existence of a rich variety of sumset patterns. As a counterpart, we stablish higher order parity obstruction to sumsets arising from nilsystems. We also provide examples of $U^k(\Phi)$-uniform sets for applications, including sets arising from the Thue-Morse and Rudin-Shapiro sequences.