 

Andrew Parrish and
Joseph Rosenblatt
Good functions for translations
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Published: 
June 9, 2022. 
Keywords: 
pointwise convergence, translation, moving derivatives, similarity conjecture. 
Subject: 
Primary 22D40, 37A15, 28D15. 


Abstract
We examine ways to describe the good functions for a.e. convergence of sequences of translations in the real line. For sequences, it is wellknown that translations are generically bad pointwise a.e., while for any integrable function there is a subsequence which is good pointwise a.e. We construct various examples of when the sequence f(x+t_{n}) does not converge a.e. or when it does converge a.e. for a sequence (t_{n}) tending to zero. In particular, for
f bounded on [0,1], we show that if for any sequence (t_{n}) tending to zero, the sequence f(x+t_{n}) converges for a.e. x, then f must be equal a.e. to a Riemann integrable function, and conversely. We discuss other techniques, issues, and questions related to sequences in the real line.


Acknowledgements
We wish to thank T. Adams for his contributions to this article, particularly in regard to Proposition 2.9, and R. Kaufman, who sent us the argument for Proposition 2.10 and an argument like the one in Proposition 2.12, giving Corollary 2.13 and Corollary 2.14. We also thank G. Edgar for many useful suggestions and A. Tserunyan for remarks on Borel, analytic, and Lebesgue sets used in the beginning of the discussion of the Erdos Similarity Problem.


Author information
Andrew Parrish:
Department of Mathematics and Computer Science
Eastern Illinois State University
600 Lincoln Avenue, Charleston, IL 619203099, USA
ajparrish@eiu.edu
Joseph Rosenblatt:
Department of Mathematics
University of Illinois at UrbanaChampaign
1409 W. Green Street, Urbana, IL 61801, USA
rosnbltt@illinois.edu

