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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Sparse generalised polynomials


Authors: Jakub Byszewski and Jakub Konieczny
Journal: Trans. Amer. Math. Soc. 370 (2018), 8081-8109
MSC (2010): Primary 37A45, 05D10, 28D05; Secondary 37B05, 11J54, 11J70, 11J71
DOI: https://doi.org/10.1090/tran/7257
Published electronically: June 26, 2018
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Abstract: We investigate generalised polynomials (i.e., polynomial-like expressions involving the use of the floor function) which take the value 0 on all integers except for a set of density 0.

Our main result is that the set of integers where a sparse generalised polynomial takes nonzero value cannot contain a translate of an IP set. We also study some explicit constructions and show that the characteristic functions of the Fibonacci and Tribonacci numbers are given by generalised polynomials. Finally, we show that any sufficiently sparse $ \{0,1\}$-valued sequence is given by a generalised polynomial.


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Additional Information

Jakub Byszewski
Affiliation: Department of Mathematics and Computer Science, Institute of Mathematics, Jagiellonian University, ul. prof. Stanisława Łojasiewicza 6, 30-348 Kraków, Poland
Email: jakub.byszewski@gmail.com

Jakub Konieczny
Affiliation: Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom
Email: jakub.konieczny@gmail.com

DOI: https://doi.org/10.1090/tran/7257
Received by editor(s): November 28, 2016
Received by editor(s) in revised form: March 2, 2017
Published electronically: June 26, 2018
Additional Notes: This research was supported by the National Science Centre, Poland (NCN), under grant no. DEC-2012/07/E/ST1/00185.
The second author also acknowledges the generous support from the Clarendon Fund and SJC Kendrew Fund for his doctoral studies.
Article copyright: © Copyright 2018 American Mathematical Society

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