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The Postage Stamp Problem and Essential Subsets in Integer Bases

  • Peter HegartyEmail author
Chapter

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 \(E(h,k) \sim (h - 1){ \log k \over \log \log k}\).

Keywords

Additive basis Essential subset 

Notes

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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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Mathematical SciencesChalmers University of Technology and Göteborg UniversityGöteborgSweden

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