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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Sparse generalised polynomials
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by Jakub Byszewski and Jakub Konieczny PDF
Trans. Amer. Math. Soc. 370 (2018), 8081-8109 Request permission

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
  • MR Author ID: 799547
  • 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
  • MR Author ID: 1178795
  • Email: jakub.konieczny@gmail.com
  • 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.
  • © Copyright 2018 American Mathematical Society
  • 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
  • MathSciNet review: 3852458