arXiv:2610.22076v1
On the structure of polynomial multicorrelation sequences arising from nilpotent actions
math.DSmath.CO37A0537A1537A3011B30
Abstract
We prove that every polynomial multicorrelation sequence arising from nilpotent actions admits a decomposition into a generalized nilsequence and a null-sequence. This resolves a long-standing question of Frantzikinakis concerning the structure of polynomial multicorrelation sequences of commuting transformations. Our proof builds upon ideas underlying the norm convergence theorem of Walsh.
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01Statements1 reported findingNo statement issues found
Every polynomial multicorrelation sequence arising from a nilpotent group action admits an exact decomposition into a generalized nilsequence and a null-sequence, with the nilsequence bounded by the original correlation bound. The commuting-transformation case follows as a special case.
Theorem 1.1No issue found
The complexity induction yields an exact nil-plus-null decomposition
PDF pages 3 and 6–10 · Section 3 · arXiv:2610.22076v1
The orthocomplement of the basic-function space makes the correlation itself null. A basic terminal function is uniformly approximated by finite interval averages, and substitution rewrites its correlation using Walsh reductions of strictly lower complexity. Induction decomposes those reduced correlations; an approximate nil-plus-small decomposition of the scalar weight then leaves a generalized nilsequence, a null term, and an arbitrarily small uniform error. The closure lemma removes that final error.
02Proofs2 reported findingsNo proof issues found
The weighted uniform-convergence proposition and the induction on Walsh complexity were checked, including the noncommutative order of the reduced polynomial sequences. Null terms remain null under bounded weights and finite averages, and products and finite sums of generalized nilsequences stay in the required class. No incorrect or materially incomplete proof was found.
Proposition 2.2No issue found in the reviewed proof
Multicorrelation weights preserve uniform convergence of polynomial averages
PDF pages 4–5 · Proposition 2.2 · arXiv:2610.22076v1
After expanding the square norm, Cauchy–Schwarz embeds the weighted difference average into an unweighted polynomial average on the product system. The combined sequences remain polynomial in the nilpotent group , so Walsh's convergence theorem makes the averages Cauchy along every Følner sequence and hence uniformly convergent.
Section 3No issue found in the reviewed proof
The lower-complexity reduction and approximation step close
PDF pages 6–10 · proof of Theorem 1.1 · arXiv:2610.22076v1
Applying to the shifted basic-function approximation produces exactly the sequences and in the Walsh reduction, whose complexity is smaller. The approximate decomposition of the remaining multicorrelation weight is uniform on all sufficiently long intervals; Cauchy–Schwarz turns its null part into an arbitrarily small uniform error. Proposition 2.3 then supplies an exact decomposition, and the nilsequence norm lemma gives the advertised bound one.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.