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

Tristán Radić

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

Dynamical Systems 2026-04-20 v2 Combinatorics Number Theory

Authors: Tristán Radić

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.

Cite

@article{arxiv.2601.06915,
  title  = {Infinite sumsets in $U^k(\Phi)$-uniform sets},
  author = {Tristán Radić},
  journal= {arXiv preprint arXiv:2601.06915},
  year   = {2026}
}

Comments

Correction in main theorem. New results added on sumsets for sets arising from constant-length substitutions, including Thue-Morse and Rudin-Shapiro applications

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

Abstract

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