Weak mixing implies weak mixing of higher orders along tempered functions

Abstract

We extend the weakly mixing PET (polynomial ergodic theorem) obtained in [2] to much wider families of functions. Besides throwing new light on the question of "how much higher degree mixing is hidden in weak mixing", the obtained results also show the way to possible new extensions of the polynomial Szemerédi theorem obtained in [6].

Single and multiple recurrence along non-polynomial sequences

“…and in particular the limit exists, proving part (1). 5 The fact that Y is a subnilmanifold of X follows from [28] or [36].…”

“…In [5,Theorem 7.1] a mean ergodic theorem along tempered sequences 2 is proved, which implies ( [5,Corollary 7.2]) that for any tempered function f , any invertible probability measure preserving system (X, B, µ, T ) and any A ∈ B, This gives a large class of sequences f (n) for which the set of optimal return times (1.1) R ε,A := n ∈ N : µ(A ∩ T −⌊f (n)⌋ A) > µ 2 (A) − ε has positive upper density. Examples include f (n) = n c (cos(log r (n)) + 2), where c > 0 and 0 < r < 1, and f (n) = bn c log r (n), where b ∈ R\{0}, c > 0 with c / ∈ N and r 0.…”

A Hardy field extension of Szemerédi's theorem

“…The ergodic method used to prove Szemerédi's theorem [24] and its polynomial extension [8] does not seem to apply, 5 so we use a different method instead. Our argument splits into three parts:…”

“…4 A result mentioned in [16] suggests the possibility that for every a ∈ H of super-polynomial growth there exists b ∈ H of the same growth, that is, the limit of b/a is a non-zero real constant, such that b(n) is an odd integer for every n ∈ N. If this is the case, then no growth assumption on elements of H with super-polynomial growth will be sufficient for our purposes. 5 The main problem appears when one deals with distal systems. Unlike the case of a polynomial with zero constant term, for a ∈ H satisfying x k ≺ a(x) ≺ x k+1 for some non-negative integer k, successive applications of the operation [a(n + m)] − [a(n)] − [a(m)], m ∈ N, lead eventually to non-zero constant sequences (in n) which is a problem when one tries to prove the corresponding coloristic (van der Waerden type) result.…”

Optimal lower bounds for multiple recurrence

“…, f k } of functions f i : N → Z. In most cases where recurrence has been established, optimal recurrence can be obtained for weakly mixing systems (see [2] when the f i are polynomials and [3,11,12] for more general f i ), or when the functions are "independent" (see [15,16] for the case of linearly independent polynomials and [12] for more general f i with different growth). In the general case, besides the aforementioned paper [4], the main progress was obtained by Frantzikinakis in [9], where the case k ≤ 3 and the f i are polynomials is studied in detail.…”