Representation functions with different weights

Quan-Hui Yang

Quan-Hui Yang

Colloquium Mathematicae, Tome 135 (2014), p. 1-6 / Harvested from The Polish Digital Mathematics Library

Résumé

For any given positive integer k, and any set A of nonnegative integers, let $r_{1, k} (A, n)$ denote the number of solutions of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. We prove that if k,l are multiplicatively independent integers, i.e., log k/log l is irrational, then there does not exist any set A ⊆ ℕ such that both $r_{1, k} (A, n) = r_{1, k} (ℕ ∖ A, n)$ and $r_{1, l} (A, n) = r_{1, l} (ℕ ∖ A, n)$ hold for all n ≥ n₀. We also pose a conjecture and two problems for further research.

Publié le : 2014-01-01

Zbl 06356981

EUDML-ID : urn:eudml:doc:283420

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     author = {Quan-Hui Yang},
     title = {Representation functions with different weights},
     journal = {Colloquium Mathematicae},
     volume = {135},
     year = {2014},
     pages = {1-6},
     zbl = {06356981},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm137-1-1}
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Quan-Hui Yang. Representation functions with different weights. Colloquium Mathematicae, Tome 135 (2014) pp. 1-6. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm137-1-1/