Multidimensional Szemerédi Theorem for Pairwise Independent Jointly Intersective Polynomials

Source 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.