PreprintCC BY 4.0arXiv
A problem of Yang and Chen on weighted representation functions
Abstract
Let ℕ denote the set of nonnegative integers. For an integer k>1 and a set A⊆ℕ, let R1,k(A,n) denote the number of solutions of n=a1+ka2 with a1,a2∈ A. For integers k>1 and t≥1, Yang and Chen defined fk(t) to be the number of sets A⊆ℕ for which R1,k(A,n)=R1,k(ℕ∖ A,n) for all integers n≥ t, and asked whether fk(t) and fl(t) are eventually equal for any integers k,l>1. We prove that fk(t)asympk 2t/tk/2, which gives a negative answer to their problem.
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BibTeX
@misc{li2026problem,
title = {{A problem of Yang and Chen on weighted representation functions}},
author = {Shuang-Shuang Li and Ya-Ting Xu and Xiao-Hui Yan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20385},
eprint = {2609.20385},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20385},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A problem of Yang and Chen on weighted representation functions
AU - Shuang-Shuang Li
AU - Ya-Ting Xu
AU - Xiao-Hui Yan
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20385
UR - https://arxiv.org/abs/2609.20385
AB - Let ℕ denote the set of nonnegative integers. For an integer k>1 and a set A⊆ℕ, let R_1,k(A,n) denote the number of solutions of n=a_1+ka_2 with a_1,a_2∈ A. For integers k>1 and t≥1, Yang and Chen defined f_k(t) to be the number of sets A⊆ℕ for which R_1,k(A,n)=R_1,k(ℕ∖ A,n) for all integers n≥ t, and asked whether f_k(t) and f_l(t) are eventually equal for any integers k,l>1. We prove that f_k(t)asymp_k 2^t/t^k/2, which gives a negative answer to their problem.
ER -