Pointwise convergence of ergodic averages with Möbius weight

2024/01/06 by Teräväinen, Joni · 2 citations
#11B30 #37A30 #37A44 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2401.03174

Abstract

Let (X,ν,T) be a measure-preserving system, and let P₁,…, P_k be polynomials with integer coefficients. We prove that, for any f₁,…, f_k∈ L^∞(X), the Möbius-weighted polynomial multiple ergodic averages (1)/(N)∑_{n≤ N}μ(n)f₁(T^P₁(n)x)⋯ f_k(T^P_k(n)x) converge to 0 pointwise almost everywhere. Specialising to P₁(y)=y, P₂(y)=2y, this solves a problem of Frantzikinakis. We also prove pointwise convergence for a more general class of multiplicative weights for multiple ergodic averages involving distinct degree polynomials. For the proofs we establish some quantitative generalised von Neumann theorems for polynomial configurations that are of independent interest.

Cited by

• On weighted multilinear polynomial averages in finite fields
• Pointwise convergence of polynomial multiple ergodic averages along the primes

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