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ON m-COVERS AND m-SYSTEMS

Part of: Diophantine equations Elementary number theory Sequences and sets

Published online by Cambridge University Press:  05 October 2009

ZHI-WEI SUN
ZHI-WEI SUN*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China (email: zwsun@nju.edu.cn)
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Abstract

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Let 𝒜={as(mod ns)}ks=0 be a system of residue classes. With the help of cyclotomic fields we obtain a theorem which unifies several previously known results related to the covering multiplicity of 𝒜. In particular, we show that if every integer lies in more than m0=⌊∑ ks=11/ns⌋ members of 𝒜, then for any a=0,1,2,… there are at least subsets I of {1,…,k} with ∑ s∈I1/ns=a/n0. We also characterize when any integer lies in at most m members of 𝒜, where m is a fixed positive integer.

Keywords

covers of the integersresidue classesunit fractions

MSC classification

Secondary: 11B25: Arithmetic progressions 11A07: Congruences; primitive roots; residue systems 11B75: Other combinatorial number theory 11D68: Rational numbers as sums of fractions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 2 , April 2010 , pp. 223 - 235
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

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