Intersective polynomials and the polynomial Szemerédi theorem
نویسندگان
چکیده
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, meaning that for any k ∈ N there exists n ∈ Zm such that the integers p1(n), . . . , pr(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 p1, . . . , pr ∈ Q[n] are jointly intersective integral polynomials, then for any finite partition Z = k i=1 Ei of Z, there exist i ∈ {1, . . . , k} and a, n ∈ Ei such that {a, a+p1(n), . . . , a+pr(n)} ⊂ Ei.
منابع مشابه
The Polynomial Multidimensional Szemerédi Theorem along Shifted Primes
If ~q1, . . . , ~qm : Z → Z are polynomials with zero constant terms and E ⊂ Z has positive upper Banach density, then we show that the set E ∩ (E − ~q1(p− 1))∩ . . .∩(E−~qm(p−1)) is nonempty for some prime p. We also prove mean convergence for the associated averages along the prime numbers, conditional to analogous convergence results along the full integers. This generalizes earlier results ...
متن کاملExtension of the Douady-Hubbard's Theorem on Connectedness of the Mandelbrot Set to Symmetric Polynimials
متن کامل
Products Of Quadratic Polynomials With Roots Modulo Any Integer
We classify products of three quadratic polynomials, each irreducible over Q, which are solvable modulo m for every integer m > 1 but have no roots over the rational numbers. Polynomials with this property are known as intersective polynomials. We use Hensel’s Lemma and a refined version of Hensel’s Lemma to complete the proof. Mathematics Subject Classification: 11R09
متن کاملA Quantitative Result on Diophantine Approximation for Intersective Polynomials
In this short note, we closely follow the approach of Green and Tao to extend the best known bound for recurrence modulo 1 from squares to the largest possible class of polynomials. The paper concludes with a brief discussion of a consequence of this result for polynomial structures in sumsets and limitations of the method.
متن کاملAlgebraic adjoint of the polynomials-polynomial matrix multiplication
This paper deals with a result concerning the algebraic dual of the linear mapping defined by the multiplication of polynomial vectors by a given polynomial matrix over a commutative field
متن کامل