Representation functions with different weights

Quan-Hui Yang. "Representation functions with different weights." Colloquium Mathematicae 137.1 (2014): 1-6. <http://eudml.org/doc/283420>.

@article{Quan2014,
abstract = {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.},
author = {Quan-Hui Yang},
journal = {Colloquium Mathematicae},
keywords = {partitions; representation functions; Sárközy problem},
language = {eng},
number = {1},
pages = {1-6},
title = {Representation functions with different weights},
url = {http://eudml.org/doc/283420},
volume = {137},
year = {2014},
}

TY - JOUR
AU - Quan-Hui Yang
TI - Representation functions with different weights
JO - Colloquium Mathematicae
PY - 2014
VL - 137
IS - 1
SP - 1
EP - 6
AB - 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.
LA - eng
KW - partitions; representation functions; Sárközy problem
UR - http://eudml.org/doc/283420
ER -