Divergence of weighted square averages in L1

Abstract

We study convergence of ergodic averages along squares with polynomial weights. For a given polynomial P∈Z[⋅], consider the set of all θ∈[0,1) such that for every ergodic system (X,μ,T) there is a function f∈L1(X,μ) such that the weighted averages along squares1N∑n=1Ne(P(n)θ)Tn2f diverge on a set with positive measure. We show that this set is residual and includes the rational numbers as well as a dense set of Liouville numbers.On one hand, this extends the divergence result for unweighted averages along squares in L1 of the first author and Mauldin; on the other hand, it shows that the convergence result for linear weights for squares in Lp, p>1, due to Bourgain as well as the second author and Krause does not hold for p=1.