2020/06/05 by Vitaly Bergelson, Bergelson, Vitaly, Joel Moreira +3 · 2 citations
Mathematics · #05D10 #11B30 #28D05 #37A44 #Advanced Topology and Set Theory #Analytic Number Theory Research #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Limits and Structures in Graph Theory
paper · doi:10.48550/arxiv.2006.03558
openalex publication_date 2020/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We obtain new results pertaining to convergence and recurrence of multiple ergodic averages along functions from a Hardy field. Among other things, we confirm some of the conjectures posed by Frantzikinakis in [Fra10; Fra16] and obtain combinatorial applications which contain, as rather special cases, several previously known (polynomial and non-polynomial) extensions of Szemeredi's theorem on arithmetic progressions [BL96; BLL08; FW09; Fra10; BMR17]. One of the novel features of our results, which is not present in previous work, is that they allow for a mixture of polynomials and non-polynomial functions. As an illustration, assume fi(t)=ai,1t^ci,1+⋯+ai,dt^ci,d for ci,j>0 and ai,j∈ℝ. Then \bullet for any measure preserving system (X,B,μ,T) and h1,…,hk∈ L^∞(X), the limit limN→∞(1)/(N)∑n=1N T[f1(n)]h1⋯ T[fk(n)]hk exists in L2; \bullet for any E⊂ ℕ with d(E)>0 there are a,n∈ℕ such that \a, a+[f1(n)],…,a+[fk(n)]\⊂ E. We also show that if f1,…,fk belong to a Hardy field, have polynomial growth, and are such that no linear combination of them is a polynomial, then for any measure preserving system (X,\mathcal B,μ,T) and any A∈\mathcal B, \limsupN→∞(1)/(N)∑n=1Nμ(A∩ T-[ f1(n) ]A∩…∩ T-[fk(n)]A) ≥ μ(A)k+1.