Unique representations of integers by linear forms

2023/03/17 by Sándor Z. Kiss, Kiss, Sándor Z., Csaba Sándor +1 · 1 citation
Mathematics · #Advanced Optimization Algorithms Research #FOS: Mathematics #Functional Equations Stability Results #Graph theory and applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2303.09878

openalex publication_date 2023/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k≥ 2 be an integer and let A be a set of nonnegative integers. For a k-tuple of positive integers \underlineλ = (λ₁, … ,λ_k) with 1 ≤ λ₁ < λ₂ < … < λ_k, we define the additive representation function R_{A,\underlineλ}(n) = |\(a₁, … ,a_k)∈ A^k: λ₁a₁ + … + λ_ka_k = n\|. For k = 2, Moser constructed a set A of nonnegative integers such that R_{A,\underlineλ}(n) = 1 holds for every nonnegative integer n. In this paper we characterize all the k-tuples \underlineλ and the sets A of nonnegative integers with R_{A,\underlineλ}(n) = 1 for every integer n≥ 0.

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• Identical representation functions of linear forms

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