Abstract
Let A be a set of integers, h ≥ 2 an integer. Let hA denote the set of all sums of h elements of A. If hA contains all sufficiently large integers, then A is called an asymptotic basis of order h. An asymptotic basis A of order h is said to be minimal if it contains no proper subset which is again an asymptotic basis of order h. This concept of minimality of bases was first introduced by Stöhr [5]. Härtter [1] showed the existence of minimal asymptotic bases by a nonconstructive argument. Nathanson [3] constructed the first nontrivial example of minimal asymptotic bases of order h ≥ 2. Jia and Nathanson [2] recently discovered a simple construction of minimal asymptotic bases of order h ≥ 2 by using powers of 2. Furthermore, for any α: 1/h ≤; α < 1, they constructed a minimal asymptotic basis A of order h such that x α < A(x) < x α, where A(x) is the number of positive elements not exceeding x. In the present paper, we shall generalize these results to g-adic representations of integers.
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References
E. Härtter, Ein Beitrag zur Theorie der Minimalbasen, J. rein angew. Math., 196(1956), 170–204.
X.-D. Jia and M. B. Nathanson, A simple construction of minimal asymptotic bases, Acta Arith., 52(1989), 95–101.
M. B. Nathanson, Minimal bases and maximal nonbases in additive number theory, J. Number Theory, 6(1974), 324–333.
M. B. Nathanson, Minimal bases and powers of 2, Acta Arith., 51(1988), 95–102.
S. A. Stöhr, Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe, II, J. reine angew. Math., 194(1955), 111–140.
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© 1996 Springer-Verlag New York, Inc.
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Jia, XD. (1996). Minimal Bases and g-Adic Representations of Integers. In: Chudnovsky, D.V., Chudnovsky, G.V., Nathanson, M.B. (eds) Number Theory: New York Seminar 1991–1995. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2418-1_15
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DOI: https://doi.org/10.1007/978-1-4612-2418-1_15
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