arXiv:2610.18657v1
Multidimensional Szemerédi Theorem for Pairwise Independent Jointly Intersective Polynomials
math.DSmath.COmath.NT37A4428D0505D1011B30
Abstract
We develop a new approach to the Joint Intersectivity Conjecture of Bergelson, Leibman, and Lesigne for commuting transformations. The conjecture asserts that the necessary divisibility condition is the only obstruction to polynomial multiple recurrence and, consequently, to the multidimensional polynomial Szemerédi theorem. We establish the conjecture for pairwise independent polynomials, obtaining the first general result for sequences of arbitrary length beyond the classical one-transformation setting. Our argument is inspired by the Furstenberg-Katznelson approach and its polynomial enhancement due to Bergelson-Leibman. Its main novelty, however, lies in the development of new techniques that allow us to bypass the use of the polynomial van der Waerden theorem, on which these methods traditionally rely, and for which no suitable analogue is currently available in our setting. In particular, we prove a Nil-Bohr returns theorem for compact extensions in systems with nilpotent structure, as well as a nil-plus-null decomposition for polynomial multicorrelation sequences that preserves joint intersectivity and positivity.
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01Statements2 reported findingsNo statement issues found
The positive solution of the joint-intersectivity conjecture for pairwise independent polynomials is stated with the hypotheses actually used, and its weighted, characteristic-factor, nil-Bohr, and compact-extension components align with the proof.
Theorems A and A-starNo issue found
The multidimensional recurrence conclusion follows from the stated polynomial assumptions
PDF pages 3–6 and 76–81 · arXiv:2610.18657v1
Joint intersectivity supplies simultaneous congruence returns, while pairwise independence is exactly the hypothesis used by the characteristic-factor estimate. The weighted ergodic statement implies the set-theoretic Szemeredi conclusion through the standard correspondence principle, with no stronger independence condition silently introduced.
Theorems 2.3 and 2.5No issue found
The nilsystem recurrence inputs are formulated at the needed generality
PDF pages 6–10 and 24–68 · arXiv:2610.18657v1
The nil-plus-null decomposition controls polynomial correlations for jointly intersective tuples, and the nil-Bohr return theorem applies to compact extensions of possibly nonergodic nilbundle factors. These statements are then used in precisely those respective roles in the final lifting argument.
02Proofs3 reported findingsNo proof issues found
Despite the length and several measurable-structure layers, the nilmanifold orbit analysis, denominator-killing construction, characteristic-factor reduction, and final compact lifting are logically connected and complete.
Section 4No issue found in the reviewed proof
Generic polynomial orbits give the required nil-plus-null decomposition
PDF pages 24–36 · arXiv:2610.18657v1
The algebraic exceptional sets are null, generic joint orbit closures are finite unions of rational subnilmanifolds, and the Goursat-type lemma identifies the connected joining from its coordinate projections and difference structure. This gives the correct limiting conditional expectation. The measurable selection step then promotes the fibrewise construction to the nonergodic setting without choosing nonmeasurable representatives.
Sections 5–6No issue found in the reviewed proof
Compact extensions and denominator killing are handled coherently
PDF pages 36–68 · arXiv:2610.18657v1
Centrality of compact structure groups is derived from the relative weak-mixing criterion, and rational lattices are enlarged compatibly along inverse systems. The decreasing-subgroup construction stabilizes the common intersection data. In the nonergodic nilbundle version, countably valued measurable lattice choices and factor gluing preserve those inclusions, yielding the stated nil-Bohr return theorem.
Sections 7–9No issue found in the reviewed proof
Characteristic factors and the final induction close without a circular step
PDF pages 69–81 · arXiv:2610.18657v1
The PET estimate removes all components orthogonal to a uniform Host–Kra factor for the pairwise independent family. Compact lifting then combines the nilsequence approximation of the factor term with joint intersectivity of the original polynomials; repeated polynomial entries occur in different roles and do not assert pairwise independence of an enlarged family. The final induction alternates this lifting with the structured recurrence bound and terminates at the announced positive lower density.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.