Partitions of Zm with the same weighted representation functions
نویسنده
چکیده
Let k = (k1, k2, · · · , kt) be a t-tuple of integers, and m be a positive integer. For a subset A ⊂ Zm and any n ∈ Zm, let rk A(n) denote the number of solutions of the equation k1a1 + · · ·+ ktat = n with a1, · · · , at ∈ A. In this paper, we give a necessary and sufficient condition on (k,m) such that there exists a subset A ⊂ Zm satisifying rk A = r k Zm\A. This settles a problem of Yang and Chen.
منابع مشابه
Partitions of ℤm with the same representation functions
For A ⊆ Zm and n ∈ Zm, let σ1(A, n), σ2(A, n), σ3(A, n) denote the number of solutions of the equation n = a+a′ with ordered pairs (a, a′) ∈ A, unordered pairs (a, a′) ∈ A(a = a′) and unordered pairs (a, a′) ∈ A, respectively. In this paper, for any i ∈ {1, 2, 3}, we determine all subsets A of Zm such that σi(A, n) = σi(Zm\A, n) holds for all n ∈ Zm.
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تاریخ انتشار 2014