Universally L^1-Bad Arithmetic Sequences

TL;DR

This paper proves that various classical sequences, including squares, powers, and primes, are universally L^1-bad, meaning they exhibit a certain divergence property in ergodic averages, extending prior results to a broader class.

Contribution

It generalizes the concept of universal L^1-badness to a wide class of sequences, including powers and primes, using a modified proof technique.

Findings

Squares, powers, and primes are universally L^1-bad.

Any subsequence of averages along these sequences retains the L^1-bad property.

The result broadens the understanding of divergence in ergodic averages for classical sequences.

Abstract

We present a modified version of Buczolich and Mauldin's proof that the sequence of square numbers is universally L^1-bad. We extend this result to a large class of sequences, including the dth powers and the set of primes; furthermore, we show that any subsequence of the averages taken along these sequences is also universally L^1-bad.