PreprintCC BY 4.0arXiv
On a conjecture on Romanoff type representations
Abstract
Let r1,…,rt be positive integers, and let 𝒜 be the set of integers of the form 2k1r1+⋯+2ktrt, where k1,…,kt are positive integers. For any positive integer n, let r(n) be the number of representations n=p+a with p an odd prime and a∈ 𝒜. We prove that if r1−1+⋯+rt−1=1, then, for every fixed positive integer m, the set of integers n with r(n)≥ m has positive lower asymptotic density. In particular, this confirms a conjecture of Chen.
Related
- From Bowditch question to Goldman conjecture for type-preserving representations
- A necessary condition for a Romanoff type theorem
- Romanoff type theorems for polynomials over finite fields
- A Romanoff-type theorem for a multiset of products of powers
- Reducibility of linear representations, free ideals, and Kippenhahn's conjecture
BibTeX
@misc{yan2026conjecture,
title = {{On a conjecture on Romanoff type representations}},
author = {Xiao-Hui Yan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13691},
eprint = {2610.13691},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13691},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On a conjecture on Romanoff type representations
AU - Xiao-Hui Yan
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13691
UR - https://arxiv.org/abs/2610.13691
AB - Let r_1,…,r_t be positive integers, and let 𝒜 be the set of integers of the form 2^k_1^r_1+⋯+2^k_t^r_t, where k_1,…,k_t are positive integers. For any positive integer n, let r(n) be the number of representations n=p+a with p an odd prime and a∈ 𝒜. We prove that if r_1^−1+⋯+r_t^−1=1, then, for every fixed positive integer m, the set of integers n with r(n)≥ m has positive lower asymptotic density. In particular, this confirms a conjecture of Chen.
ER -