On the supremum of the representation function of a sumset
On the supremum of the representation function of a sumset
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DOI：
10.2989/16073606.2013.779961
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发表时间：
2014-01
影响因子：
0.7
通讯作者：
A. Dubickas
中科院分区：
文献类型：
--
作者：
A. Dubickas
摘要：设A为非负整数集合∧∪{0}的一个子集，设rA (n)为n≥0的A + b与A， b∈A的和表示的个数。定义s(A):= supn≥0 rA (n)对于每个A∈A∈A。给定任意三个数a， b, c∈{2,3,4，…}∪{∞}满足max(a, b)≤c，证明了N存在两个无限不相交的子集a， b，使得对于每个N∈N， a的第N个元素不超过b的第N个元素，s(a) = a, s(b) = b和s(a∪b) = c。这推广了Grekos， Haddad， Helou和Pihko的两个结果。
Abstract Let A be a subset of the set of nonnegative integers ℕ ∪ {0}, and let rA (n) be the number of representations of n ≥ 0 by the sum a + b with a, b ∈ A. Define s(A):= supn≥0 r A (n) for each A ⊆ ℕ ∪ {0}. Given any three numbers a, b, c ∈ {2, 3, 4,…} ∪ {∞} satisfying max(a, b) ≤ c we prove that there exist two infinite disjoint subsets A, B of N such that for each n ∈ ℕ the nth element of A does not exceed the nth element of B, s(A) = a, s(B) = b and s(A ∪ B) = c. This generalizes two results of Grekos, Haddad, Helou and Pihko.