Intersective polynomials and the polynomial Szemerédi theorem

Abstract

Let P = { p 1 , … , p r } ⊂ Q [ n 1 , … , n m ] be a family of polynomials such that p i ( Z m ) ⊆ Z , i = 1 , … , r . We say that the family P has the PSZ property if for any set E ⊆ Z with d ∗ ( E ) = lim sup N − M → ∞ | E ∩ [ M , N − 1 ] | N − M > 0 there exist infinitely many n ∈ Z m such that E contains a polynomial progression of the form { a , a + p 1 ( n ) , … , a + p r ( n ) } . We prove that a polynomial family P = { p 1 , … , p r } has the PSZ property if and only if the polynomials p 1 , … , p r are jointly intersective, meaning that for any k ∈ N there exists n ∈ Z m such that the integers p 1 ( n ) , … , p r ( n ) are all divisible by k. To obtain this result we give a new ergodic proof of the polynomial Szemerédi theorem, based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds. We also obtain, as a corollary, the following generalization of the polynomial van der Waerden theorem: If p 1 , … , p r ∈ Q [ n ] are jointly intersective integral polynomials, then for any finite partition Z = ⋃ i = 1 k E i of Z , there exist i ∈ { 1 , … , k } and a , n ∈ E i such that { a , a + p 1 ( n ) , … , a + p r ( n ) } ⊂ E i .