A new mathematical preprint reports that a set with “nice recurrence” cannot be broken into finitely many colour classes without at least one class keeping that same property. In the language of the paper, the result establishes partition regularity: every finite partition of a nice-recurrence set contains a part that is itself nice recurrence.
The finding addresses whether every set of nice recurrence has this Ramsey-style behaviour under finite colouring. It is a proof about abstract mathematical objects, rather than a result from observations, measurements or a sampled population.
What “nice recurrence” means
The term describes a form of return behaviour. A set R has nice recurrence when, in every measure-preserving system, and for every measurable set and every positive tolerance, at least one n in R gives a return intersection at least as large as the squared measure of the set minus that tolerance. Put more plainly, the permitted return times in R must come close to the level expected from the size of the set, across all systems covered by the definition.
The systems in the proof are abstract probability-measure spaces equipped with measurable maps that preserve measure. The analysis involves measurable sets and functions inside those systems; there are no human participants, animals, laboratory treatments or empirical data behind the theorem.
That distinction matters because the paper’s conclusion is existential. It guarantees that one colour class has nice recurrence, but it does not say which class it is, nor does it give a numerical estimate for how large that class must be.
A proof built around failure
The argument proceeds by contraposition, a way of proving a statement by showing that its failure would force the opposite conclusion. It also uses induction on the number of colours. The key step is to show that if two sets both fail to be nice recurrence, then their union fails as well; repeating that step supports the result for any finite partition.
To describe that failure precisely, the paper gives an if-and-only-if functional test. A set is not nice recurrence exactly when there is a bounded measurable real-valued function with mean zero whose correlations remain uniformly negative over the set. A correlation here is an integral that measures how the function aligns with a shifted copy of itself.
The proof starts with the two functions that witness failure for the two component sets. It then places the corresponding systems together in a larger product measure-preserving system. This construction lets the two separate witnesses be examined at the same return time while retaining the measure structure required by the definition.
The combined witness is controlled through a polynomial whose coefficients are nonnegative. The polynomial lemma shows that whenever at least one of its input correlations lies below a fixed negative threshold, the output is also negative and is bounded away from zero by a uniform amount. The separating constant is existential: the proof establishes that such a bound exists without evaluating it as a particular numerical value.
For the constructed witness, the correlation at a given n is represented by evaluating that polynomial at the two component correlations. Since n belongs to the union of the two non-nice-recurrence sets, at least one component supplies the required negative input. The resulting correlation is therefore bounded above by a negative constant, which is the step that shows the union is not nice recurrence.
A theorem, not a measured effect
The paper reports no statistical uncertainty because it does not estimate an effect from data. Its conclusion is conditional on the stated definitions and the validity of the proof. The main result is a theorem about sets of positive integers and finite partitions; an adaptation to arbitrary abelian groups is mentioned only as a remark.
The result stops short of identifying a rule for finding the successful colour class in a particular partition. It also provides no quantitative density bound. What it establishes is the existence of at least one qualifying class, not a guarantee that every colour class has nice recurrence.
The document is an arXiv version 1 preprint dated 20 August 2026. The author reports support from EPSRC through Joel Moreira’s Frontier Research Guarantee grant, reference EP/Y014030/1.
For readers outside the subject, the significance is structural: the proof shows that this recurrence property survives in at least one branch of every finite colouring. It does not test the result empirically or establish effects in people or animals.
Paper data and sources
Original title: Sets of nice recurrence are partition regular
Authors: Jonathan Chapman
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text