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On the maximum density of minimal asymptotic bases


Authors: Melvyn B. Nathanson and András Sárközy
Journal: Proc. Amer. Math. Soc. 105 (1989), 31-33
MSC: Primary 11B13; Secondary 11B05, 11P99
DOI: https://doi.org/10.1090/S0002-9939-1989-0973836-1
MathSciNet review: 973836
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Abstract: A 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$. It is proved that if $ A$ is an asymptotic basis of order $ h$ with lower asymptotic density $ {d_L}(A) > 1/h$, then there is a set $ W$ contained in $ A$ such that $ W$ has positive asymptotic density and $ A\backslash W$ is an asymptotic basis of order $ h$. This implies that if $ A$ is a minimal asymptotic basis of order $ h$, then $ {d_L}(A) \leq 1/h$.


References [Enhancements On Off] (What's this?)

  • [1] P. Erdös and M. B. Nathanson, Problems and results on minimal bases in additive number theory, in: Number Theory, New York 1985-86, Lecture Notes in Math., vol. 1240, Springer-Verlag, Heidelberg, 1987, pp. 87-96. MR 894505 (88j:11006)
  • [2] -, and -, Minimal asymptotic bases with prescribed densities, Illinois J. Math. 32 (1988), to appear. MR 947047 (89j:11010)
  • [3] E. Härtter, Ein Beitrag zur Theorie der Minimalbasen, J. Reine Angew. Math. 196 (1956), 170-204. MR 0086086 (19:122a)
  • [4] M. Kneser, Abschätzung der asymptotischen Dichte von Summenmengen, Math. Z. 58 (1953), 459-484. MR 0056632 (15:104c)
  • [5] M. B. Nathanson, Minimal bases and maximal nonbases in additive number theory, J. Number Theory 6 (1974), 324-333. MR 0347764 (50:265)
  • [6 -, Minimal bases and powers of 2, Acta Arith] 51 (1988), 95-102. MR 967335 (89m:11015)
  • [7] A. Stöhr, Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe. II, J. Reine Angew. Math. 194 (1955), 111-140.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0973836-1
Keywords: Minimal asymptotic bases, additive bases, sumsets, additive number theory
Article copyright: © Copyright 1989 American Mathematical Society

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