On the minimal Hamming weight of a multi-base representation
Given a finite set of bases b1, b2, …, br (integers greater than 1), a multi-base representation of an integer n is a sum with summands db1α1b2α2⋯brαr, where the αj are nonnegative integers and the digits d are taken from a fixed finite set. We consider multi-base representations with at least two b...
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| Published in | Journal of number theory Vol. 208; pp. 168 - 179 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Elsevier Inc
01.03.2020
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| ISSN | 0022-314X 1096-1658 1096-1658 |
| DOI | 10.1016/j.jnt.2019.07.023 |
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| Abstract | Given a finite set of bases b1, b2, …, br (integers greater than 1), a multi-base representation of an integer n is a sum with summands db1α1b2α2⋯brαr, where the αj are nonnegative integers and the digits d are taken from a fixed finite set. We consider multi-base representations with at least two bases that are multiplicatively independent. Our main result states that the order of magnitude of the minimal Hamming weight of an integer n, i.e., the minimal number of nonzero summands in a representation of n, is logn/(loglogn). This is independent of the number of bases, the bases themselves, and the digit set.
For the proof, the existing upper bound for prime bases is generalized to multiplicatively independent bases; for the required analysis of the natural greedy algorithm, an auxiliary result in Diophantine approximation is derived. The lower bound follows by a counting argument and alternatively by using communication complexity; thereby improving the existing bounds and closing the gap in the order of magnitude. This implies also that the greedy algorithm terminates after O(logn/loglogn) steps, and that this bound is sharp. |
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| AbstractList | Given a finite set of bases b1, b2, …, br (integers greater than 1), a multi-base representation of an integer n is a sum with summands db1α1b2α2⋯brαr, where the αj are nonnegative integers and the digits d are taken from a fixed finite set. We consider multi-base representations with at least two bases that are multiplicatively independent. Our main result states that the order of magnitude of the minimal Hamming weight of an integer n, i.e., the minimal number of nonzero summands in a representation of n, is logn/(loglogn). This is independent of the number of bases, the bases themselves, and the digit set.
For the proof, the existing upper bound for prime bases is generalized to multiplicatively independent bases; for the required analysis of the natural greedy algorithm, an auxiliary result in Diophantine approximation is derived. The lower bound follows by a counting argument and alternatively by using communication complexity; thereby improving the existing bounds and closing the gap in the order of magnitude. This implies also that the greedy algorithm terminates after O(logn/loglogn) steps, and that this bound is sharp. |
| Author | Wagner, Stephan Krenn, Daniel Suppakitpaisarn, Vorapong |
| Author_xml | – sequence: 1 givenname: Daniel surname: Krenn fullname: Krenn, Daniel email: math@danielkrenn.at, daniel.krenn@aau.at organization: Department of Mathematics, Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65–67, 9020 Klagenfurt, Austria – sequence: 2 givenname: Vorapong surname: Suppakitpaisarn fullname: Suppakitpaisarn, Vorapong email: vorapong@is.s.u-tokyo.ac.jp organization: Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan – sequence: 3 givenname: Stephan surname: Wagner fullname: Wagner, Stephan email: swagner@sun.ac.za organization: Department of Mathematical Sciences, Stellenbosch University, Private Bag X1, Matieland 7602, South Africa |
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| Keywords | 94A15 11J25 68R05 11A63 Minimal weight Multi-base representations Hamming weight |
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| Snippet | Given a finite set of bases b1, b2, …, br (integers greater than 1), a multi-base representation of an integer n is a sum with summands db1α1b2α2⋯brαr, where... |
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| StartPage | 168 |
| SubjectTerms | Hamming weight Minimal weight Multi-base representations |
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| Title | On the minimal Hamming weight of a multi-base representation |
| URI | https://dx.doi.org/10.1016/j.jnt.2019.07.023 https://doi.org/10.1016/j.jnt.2019.07.023 |
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