Polynomial extensions of van der Waerden’s and Szemerédi’s theorems

Abstract

An extension of the classical van der Waerden and Szemerédi theorems is proved for commuting operators whose exponents are polynomials. As a consequence, for example, one obtains the following result: Let S ⊆ Z l S\subseteq \mathbb {Z}^l be a set of positive upper Banach density, let p 1 ( n ) , … , p k ( n ) p_1(n),\dotsc ,p_k(n) be polynomials with rational coefficients taking integer values on the integers and satisfying p i ( 0 ) = 0 p_i(0)=0 , i = 1 , … , k ; i=1,\dotsc ,k; then for any v 1 , … , v k ∈ Z l v_1,\dotsc ,v_k\in \mathbb {Z}^l there exist an integer n n and a vector u ∈ Z l u\in \mathbb {Z}^l such that u + p i ( n ) v i ∈ S u+p_i(n)v_i\in S for each i ≤ k i\le k .

Szemerédi's theorem and problems on arithmetic progressions

“…We also note the remarkable Bergelson-Leibman theorem ( [19]; see also [71], [133]), already mentioned in the Introduction.…”

“…The inequality (18) and the equality (19) imply the existence of a progression P l0 j0 such that k∈ e P l 0 j 0…”

Теорема Семереди И Задачи Об Арифметических Прогрессиях

“…Отметим также замечательную теорему В. Бергельсона и А. Лейбмана [19] (см. также [71]), уже упоминавшуюся во введении.…”

A nilpotent IP polynomial multiple recurrence theorem

“…In a different direction, Bergelson and Leibman [BL96] have proved a polynomial multiple recurrence theorem. That result has been extended from commutative to nilpotent groups of transformations by Leibman [Lei98].…”