PreprintCC BY 4.0arXiv
Infinite prime sumsets in structured and Uk(Φ)-uniform sets
Abstract
By introducing new ergodic-theoretic techniques in nilsystems, we determine which infinite sumset configurations occur in Uk(Φ)-uniform and Nil-Bohr sets. To be more precise, our first result associates the degree k of a Uk(Φ)-uniform set with the variety of sumsets it contains, solving a conjecture of Kra, Moreira, Richter and Robertson. Restricting to Nil-Bohr sets we show the existence of infinite sumsets with summands in the shifted primes ℙ−1. As a consequence, we show that for any real polynomial Q(n) with leading irrational coefficient of degree k, and any natural numbers ℓ1, ⋯, ℓk there is an infinite set P⊂ ℙ such that Q(∑p ∈ I p) ∈ U pmod 1 for all I ⊂ P , |I| = ℓ1, …, ℓk.
Related
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- Infinite Families of Counterexamples to a Conjecture of Liu and Qian and a Refined Inverse Theorem for Restricted Sumsets in ℤp
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BibTeX
@misc{hernandez2026infinite,
title = {{Infinite prime sumsets in structured and $U^k(Φ)$-uniform sets}},
author = {Felipe Hernández and Tristán Radić},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20522},
eprint = {2609.20522},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20522},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Infinite prime sumsets in structured and U^k(Φ)-uniform sets
AU - Felipe Hernández
AU - Tristán Radić
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20522
UR - https://arxiv.org/abs/2609.20522
AB - By introducing new ergodic-theoretic techniques in nilsystems, we determine which infinite sumset configurations occur in U^k(Φ)-uniform and Nil-Bohr sets. To be more precise, our first result associates the degree k of a U^k(Φ)-uniform set with the variety of sumsets it contains, solving a conjecture of Kra, Moreira, Richter and Robertson. Restricting to Nil-Bohr sets we show the existence of infinite sumsets with summands in the shifted primes ℙ−1. As a consequence, we show that for any real polynomial Q(n) with leading irrational coefficient of degree k, and any natural numbers ℓ_1, ⋯, ℓ_k there is an infinite set P⊂ ℙ such that Q(∑_p ∈ I p) ∈ U pmod 1 for all I ⊂ P , |I| = ℓ_1, …, ℓ_k.
ER -