Abstract
The set A is an asymptotic nonbasis of order h for an additive abelian semigroup X if there are infinitely many elements of X not in the h-fold sumset hA. For all \(h \geq 2\), this paper constructs new classes of asymptotic nonbases of order h for Z and for N0 that are not subsets of maximal asymptotic nonbases.
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Deshouillers, J.-M., Grekos, G.: Propriétés extrémales de bases additives. Bull. Soc. Math. France 107, 319–335 (1979)
Erdős, P., Nathanson, M.B.: Maximal asymptotic nonbases. Proc. Amer. Math. Soc. 48, 57–60 (1975)
Erdős, P., Nathanson, M.B.: Oscillations of bases for the natural numbers. Proc. Amer. Math. Soc. 53(2), 253–258 (1975)
Erdős, P., Nathanson, M.B.: Partitions of the natural numbers into infinitely oscillating bases and nonbases. Comment. Math. Helv. 51, 171–182 (1976)
P. Erdős and M. B. Nathanson, Nonbases of density zero not contained in maximal nonbases, J. London Math. Soc. (2), 15 (1977), 403–405
Erdős, P., Nathanson, M.B.: Bases and nonbases of square-free integers. J. Number Theory 11, 197–208 (1979)
P. Erdős and M. B. Nathanson, Problems and results on minimal bases in additive number theory, in: Number Theory (New York, 1984–1985), Lecture Notes in Math., vol. 1240, Springer (Berlin, 1987), pp. 87–96
Hennefeld, J.: Asymptotic nonbases which are not subsets of maximal aymptotic nonbases. Proc. Amer. Math. Soc. 62, 23–24 (1977)
Ling, D.: A note on asymptotic nonbases. Bull. Aust. Math. Soc. 95, 1–4 (2017)
Ling, D.: A construction of maximal asymptotic nonbases. Int. J. Number Theory 14(4), 919–923 (2018)
Nathanson, M.B.: Minimal bases and maximal nonbases in additive number theory. J. Number Theory 6, 324–333 (1974)
M. B. Nathanson, s-maximal nonbases of density zero, J. London Math. Soc. (2), 15 (1977), 29–34
M. B. Nathanson, Additive problems in combinatorial number theory, in: Number Theory (New York, 1985–1988), Lecture Notes in Math., vol. 1383, Springer (Berlin, 1989), pp. 123–139
M. B. Nathanson, Problems in additive number theory. III, in: Combinatorial Number Theory and Additive Group Theory, Adv. Courses Math. CRM Barcelona, Birkhäuser Verlag (Basel, 2009), pp. 279–297
Nathanson, M.B., Sárközy, A.: Metric theorems on minimal bases and maximal nonbases. Studia Sci. Math. Hungar. 32, 207–226 (1996)
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To Endre Szemerédi on his 80th birthday
This research was supported by a grant from the PSC-CUNY Research Awards Program.
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Nathanson, M.B. Congruence classes and maximal nonbases. Acta Math. Hungar. 161, 768–779 (2020). https://doi.org/10.1007/s10474-020-01075-w
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DOI: https://doi.org/10.1007/s10474-020-01075-w