Intersective polynomials and the polynomial Szemerédi theorem

@inproceedings{Bergelson2008IntersectivePA,
  title={Intersective polynomials and the polynomial Szemer{\'e}di theorem},
  author={Vitaly Bergelson and Alexander Leibman and Emmanuel Lesigne},
  year={2008}
}
Let P = {p1, . . . , pr} ⊂ Q[n1, . . . , nm] be a family of polynomials such that pi(Z) ⊆ Z, i = 1, . . . , r. We say that the family P has the PSZ property if for any set E ⊆ Z with d∗(E) = lim supN−M→∞ |E∩[M,N−1]| N−M > 0 there exist infinitely many n ∈ Zm such that E contains a polynomial progression of the form {a, a + p1(n), . . . , a + pr(n)}. We prove that a polynomial family P = {p1, . . . , pr} has the PSZ property if and only if the polynomials p1, . . . , pr are jointly intersective… CONTINUE READING