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On finite additive $ 2$-bases

Author: Laurent Habsieger
Journal: Trans. Amer. Math. Soc. 366 (2014), 6629-6646
MSC (2010): Primary 11B13; Secondary 11B34
Published electronically: July 17, 2014
MathSciNet review: 3267021
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Abstract: For a positive integer $ N$, a set $ \mathcal {B}$ of integers from $ \{0,1,\dots ,N-1\}$ is called an additive $ 2$-basis for $ N$ if every integer $ n\in \{0,1,\dots ,N-1\}$ may be represented as the sum of $ 2$ elements of $ \mathcal {B}$. We discuss the methods used to estimate the minimal size of an additive $ 2$-basis for $ N$. We provide new examples to enrich this survey, which give good bounds. For instance, we slightly improve on the current record, from $ 0.46972$ to $ 0.46906$.

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Laurent Habsieger
Affiliation: Centre de Recherches Mathématiques, CNRS UMI 3457, Université de Montréal, Case Postale 6128, Succursale Centre-Ville, Montréal, Quebec, Canada H3C 3J7

Keywords: Additive bases
Received by editor(s): April 15, 2013
Published electronically: July 17, 2014
Additional Notes: This work was supported by the French National Agency for Research (CAESAR ANR-12-BS01-0011). The author also thanks Alain Plagne and Victor Lambert for their careful reading of a preliminary version of this paper.
Article copyright: © Copyright 2014 American Mathematical Society