Summary
Plagne recently determined the asymptotic behavior of the function E(h), which counts the maximum possible number of essential elements in an additive basis for \(\mathbb{N}\) of order h. Here, we extend his investigations by studying asymptotic behavior of the function E(h, k), which counts the maximum possible number of essential subsets of size k, in a basis of order h. For a fixed k and with h going to infinity, we show that \(E(h,k) = {\Theta }_{k}\left ({[{h}^{k}/\log h]}^{1/(k+1)}\right )\). The determination of a more precise asymptotic formula is shown to depend on the solution of the well-known ‘ postage stamp problem’ in finite cyclic groups. On the other hand, with h fixed and k going to infinity, we show that \(E(h,k) \sim (h - 1){ \log k \over \log \log k}\).
Mathematics Subject Classifications (2000). 11B13 (primary), 11B34 (secondary)
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Notes
- 1.
We have not seen the numbers defined in equations (2.4), (2.5), (2.10), (2.11), (2.17) and (2.18) introduced explicitly in the existing literature on the PSP.
- 2.
The notation d(N, k) is common in the literature, since these numbers can be interpreted as diameters of Cayley graphs.
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Acknowledgements
I thank Alain Plagne for very helpful discussions and Melvyn Nathanson for some literature tips on the PSP. This work was completed while I was visiting City University of New York, and I thank them for their hospitality. My research is partly supported by a grant from the Swedish Research Council (Vetenskapsrådet).
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Dedicated to Melvyn B. Nathanson on the occasion of his 60th birthday
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Hegarty, P. (2010). The Postage Stamp Problem and Essential Subsets in Integer Bases. In: Chudnovsky, D., Chudnovsky, G. (eds) Additive Number Theory. Springer, New York, NY. https://doi.org/10.1007/978-0-387-68361-4_11
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