Algebraic characterization of RP^[d] via new topology and proof that order d-1 maximal factors are topological characteristic factors for higher-order configurations in group actions.
Cube structures of the universal minimal system, nilsystems and applications
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abstract
We propose and develop an approach to study nilsystems and their proximal extensions using cube structures associated with the universal minimal system. We provide alternative proofs for results regarding saturation properties of factor maps to maximal nilfactors in cubes, as well as new results and applications of independent interest to the structural theory of topological systems. In particular, we give a new proof that $\mathbf{RP}^{[d]}$ is an equivalence relation, building upon the distal case, by establishing a description of this relation in algebraic terms. This is new even for d=1.
fields
math.DS 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
U^k(Φ)-uniform sets contain rich families of infinite sumsets whose structure scales with k, subject to higher-order parity obstructions coming from nilsystems.
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On higher order regionally proximal relations and topological characteristic factors for group actions
Algebraic characterization of RP^[d] via new topology and proof that order d-1 maximal factors are topological characteristic factors for higher-order configurations in group actions.
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Infinite sumsets in $U^k(\Phi)$-uniform sets
U^k(Φ)-uniform sets contain rich families of infinite sumsets whose structure scales with k, subject to higher-order parity obstructions coming from nilsystems.