On m-covers and m-systems
Abstract
Let A=as(mod ns)s=0k be a system of residue classes. With the help of cyclotomic fields we obtain a theorem which unifies several previously known results concerning system A. In particular, we show that if every integer lies in more than m=[sums=1k 1/ns] members of A, then for any a=0,1,2,... there are at least binomm[a/n0] subsets I of 1,...,k with sums in 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.
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