Acta Arith. 81(1997), no. 2, 175-198. EXACT m-COVERS AND THE LINEAR FORM ∑ k s=1 xs/ns
نویسندگان
چکیده
where N is the least common multiple of those common differences (or moduli) n1, · · · , nk. For a positive integer m, (1) is said to be an m-cover of Z if its covering multiplicity is not less than m, and an exact m-cover of Z if σ(x) = m for all x ∈ Z. Note that k > m if (1) forms an m-cover of Z. Clearly the covering function σ : Z → Z is constant if and only if (1) forms an exact m-cover of Z for some m = 1, 2, 3, · · · . An exact 1-cover of Z is a partition of Z into residue classes. P. Erdös ([E]) proposed the concept of cover (i.e., 1-cover) of Z in 1930’s, Š. Porubský ([P]) introduced the notion of exactm-cover of Z in 1970’s, and the author ([Su3]) studied m-covers of Z for the first time. The most challenging problem in this field is to describe those n1, · · · , nk in an m-cover (or exact m-cover) (1) of Z (cf. [Gu]). In [Su2,Su3,Su4] the author revealed some connections between (exact) m-covers of Z and Egyptian fractions. Here we concentrate on exact m-covers of Z. In [Su3,Su4] results for exact m-covers of Z were obtained by studying general
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