On some maximal and minimal sets

A set A of positive integers is called 3-free if it contains no 3-term arithmetic progression. Furthermore, such A is called maximal if it is not properly contained in any other 3-free set. In 2006, by confirming a question posed by Erdős et al., Savchev and Chen proved that there exists a maximal 3...

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Bibliographic Details
Published in:Periodica mathematica Hungarica 2024-06, Vol.88 (2), p.354-358
Main Authors: Fang, Jin-Hui, Cao, Xue-Qin
Format: Article
Language:English
Subjects:
Asymptotic properties
Integers
Mathematics
Mathematics and Statistics
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Summary:A set A of positive integers is called 3-free if it contains no 3-term arithmetic progression. Furthermore, such A is called maximal if it is not properly contained in any other 3-free set. In 2006, by confirming a question posed by Erdős et al., Savchev and Chen proved that there exists a maximal 3-free set { a 1 < a 2 < ⋯ < a n < ⋯ } of positive integers with the property that lim n → ∞ ( a n + 1 - a n ) = ∞ . In this paper, we generalize their result. On the other hand, a set A of nonnegative integers is called an asymptotic basis of order h if every sufficiently large integer can be represented as a sum of h elements of A . Such A is defined as minimal if no proper subset of A has this property. We also extend a result of Jańczak and Schoen about the minimal asymptotic basis.
ISSN:0031-5303
1588-2829
DOI:10.1007/s10998-023-00559-w