Journal Article10.1007/BF02568149
Partitions of the natural numbers into infinitely oscillating bases and nonbases
11
TL;DR: In this article, a partition of the natural numbers into a basisA and a nonbasisB such that, as random elements are moved one at a time from A to B, from B to A, from A-B, to B-A, from C-A to B -B, the setA oscillates from basis to non-basis to basis.
read more
Abstract: The setA of nonnegative integers is a basis if every sufficiently large integerx can be written in the formx=a+a′ witha, a′∈A. IfA is not a basis, then it is a nonbasis. We construct a partition of the natural numbers into a basisA and a nonbasisB such that, as random elements are moved one at a time fromA toB, fromB toA, fromA toB, …, the setA oscillates from basis to nonbasis to basis … and the setB oscillates simultaneously from nonbasis to basis to nonbasis…
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
Systems of distinct representatives and minimal bases in additive number theory
Paul Erdös,Melvyn B. Nathanson +1 more
- 01 Jan 1979
TL;DR: In this article, it was shown that the set A of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer is the sum of h elements of A.
Problems in additive number theory, III
TL;DR: In this article, the cardinality of a set of positive integers, non-negative integers, integers and d-dimensional integral lattice points is defined as a function of cardinality.
17
Minimal Asymptotic Bases for the Natural Numbers
TL;DR: In this article, it was shown that there exists an asymptotic basis A such that M h A (x) = O(x 1−1 h+ϵ ) for every ϵ > 0.
13
Oscillations of bases in number theory and combinatorics
Melvyn B. Nathanson
- 01 Jan 1977
TL;DR: In this article, it was shown that the set B is an asymptotic basis of order h if hB = IN, but hB' / IN for every proper subset B' ~ B.
8
Addictive Number Theory
Melvyn B. Nathanson,Melvyn B. Nathanson +1 more
- 01 Jan 2010
TL;DR: In 1996, just after Springer-Verlag published my books Additive Number Theory: The Classical Bases and Addictive Number Theory, I went into my local Barnes and Noble superstore and looked for them on the shelves, and found that the books did not exist.
5
References
Minimal bases and maximal nonbases in additive number theory
TL;DR: It is proved that every set not a basis of order h is a subset of a maximal nonbasis of order g, which is a set of nonnegative integers such that every proper superset of B is a basis.
58
Oscillations of bases for the natural numbers
Paul Erdős,Melvyn B. Nathanson +1 more
- 01 Feb 1975
TL;DR: In this paper, a set of positive integers is defined as a basis if every sufficiently large integer n can be written in the form n = ai + a; with a., aI e A.
20
Maximal asymptotic nonbases
Paul Erdős,Melvyn B. Nathanson +1 more
- 01 Jan 1975
TL;DR: In this article, it was shown that any maximal asymptotic nonbasis of order 2 that satisfies a certain finiteness condition is a subset of a maximal non-congruence non-basis.
Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe. I.
TL;DR: In § l definierten xS-Basen genauer untersucht werden as discussed by the authors, wenn eine natürliche Zahl m and eine unendliche arithmetische Progression s, s + i, s+ 2i,... with s > 0 and t > 0 so gibt, daß jede Zahl dieser Progression Summe von höchstens m Zahlen aus 9l ist.
Related Papers (5)
Paul Erdős,Melvyn B. Nathanson +1 more
- 01 Feb 1975
Paul Erdős,Melvyn B. Nathanson +1 more
- 01 Jan 1975