Divergent square averages

By Z. Buczolich, R. Daniel Mauldin. 2005

72 citations recorded by the source corpus.

In this paper we answer a question of J. Bourgain which was motivated by questions A. Bellow and H. Furstenberg. We show that the sequencefn 2 g 1D1 is L 1 -universally bad. This implies that it is not true that given a dynamical system .X;U;;T/ andf 2L 1 ./ , the ergodic means

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4f6dd6554f52eb71a586c5ff44f78486856c6797

How researchers see this paper

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In Algebra and Number Theory

1. Mariusz Mirek
2. Terence Tao
3. Ben Krause
4. Nikos Frantzikinakis
5. B. Host
6. Ben Joseph Green
7. T. Ziegler
8. Dorin Ervin Dutkay

In Discrete Mathematics and Combinatorics

1. Mariusz Mirek
2. Terence Tao
3. Ben Krause
4. Nikos Frantzikinakis
5. Van H. Vu
6. B. Host
7. Charles Bordenave
8. L. Erdős

In Mathematical Physics

1. Mariusz Mirek
2. Terence Tao
3. Ben Krause
4. Bartosz Trojan
5. Van H. Vu
6. Zihua Guo
7. D. Tataru
8. Carlos E. Kenig

Citations in the corpus

• Discrete maximal functions in higher dimensions and applications to ergodic theory (2014, background): “Recently, Buczolich and Mauldin [5] have shown that the pointwise convergence of A N f fails on L 1 ( X,μ ) (see also [20]).”
• Pointwise ergodic theorems for non-conventional bilinear polynomial averages (2020, methodology): “is also shown that (1.6) fails at the endpoint p= r= 2. For p= 1, in contrast to Theorem1.4(ii), pointwise convergence in Theorem1.5(ii) fails for any monomial P(n) = ndof degree d2, as was shown in [16,49]. Theorem1.5is proven via the circle method. The implementation of this method can be summariz”