A Note on a problem of Sarkozy and Sos
A Note on a problem of Sarkozy and Sos
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DOI：
10.3770/j.issn:2095-2651.2022.03.003
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Min Tang
中科院分区：
文献类型：
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作者：
Min Tang
Let k, ℓ ≥ 2 be positive integers. Let A be an infinite set of nonnegative integers. For n ∈ N, let r_(1,k,...,k~(ℓ-1))(A, n) denote the number of solutions of n = a_0 +ka_1 +···+k~(ℓ-1)a_(ℓ-1), a_0,..., a_(ℓ-1) ∈ A. In this paper, we show that r_(1,k,...,k~(ℓ-1))(A, n) = 1 for all n ≥ 0 if and only if A is the set of all nonnegative integers such that all its digits in its k~ℓ-adic expansion are smaller than k. This result partially answers a question of Sarkozy and Sos on representation for multivariate linear forms.
