Divergence of weighted square averages in L1

We study convergence of ergodic averages along squares with polynomial weights. For a given polynomial P is an element of Z[center dot], consider the set of all theta is an element of [0, 1) such that for every ergodic system ( X, mu, T) there is a function f is an element of L-1( X, mu) such that the weighted averages along squares 1/N Sigma(N)(n=1)e(P(n)theta)Tn(2) f 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 L-1 of the first author and Mauldin; on the other hand, it shows that the convergence result for linear weights for squares in L-p, p > 1, due to Bourgain as well as the second author and Krause does not hold for p = 1. (C) 2021 The Author(s). Published by Elsevier Inc.