Single and multiple recurrence along non-polynomial sequences

Abstract

We establish new recurrence and multiple recurrence results for a rather large

Furstenberg Systems of Bounded Multiplicative Functions and Applications

“…For example if w(n) = n, n ∈ N, then we get the Cèsaro averages, and if w(n) = log n, n ∈ N, then we get an averaging scheme equivalent to logarithmic averages. The next result is a direct consequence of results proved in [7].…”

“…Instead, we proceed by showing in Theorem 5.1 that the Furstenberg systems of such sequences are strongly stationary (a property that fails when the sequence (a(n)) is polynomial). The proof of this fact follows from the multiple ergodic theorem of Proposition 5.7, which in turn is proved using recent deep results of Bergelson, Moreira, and Richter [7], using the theory of characteristic factors of Host-Kra [26] and equidistribution results on nilmanifolds. The structure of strongly stationary systems was determined in [20,28] and we use these structural results as a black box in order to complete the proof of Theorem 1.6.…”

“…To prove this, two modifications are needed in our argument: The first is in the definition of Furstenberg systems, one has to use the weighted averages E w n∈N for w = a (d) (see Section 5.1 for their definition) in place of the usual Cèsaro averages. The second is in Section 5.1 one has to use the corresponding equidistribution results from [7] for the weighted averages. The same comment applies for Corollary 1.7.…”

“…The proof of Theorem 5.2 will be based on an equidistribution result from [7] that was proved for certain weighted averages that we define next. For r ∈ N, let ∆ 1 r a = a(n + r) − a(n), n ∈ N, and for i ∈ N define inductively ∆ i+1 r a = ∆ r (∆ i r a).…”

“…Strong stationarity of Hardy field iterates. We will use the following result that follows from Theorem D and Theorem 4.5 in [7]: Theorem 5.6 ( [7]). Let a : R + → R be a Hardy field function such that t d+ε ≺ a(t) ≺ t d+1 for some d ∈ Z + and ε > 0.…”