PreprintCC BY 4.0arXiv
A necessary condition for a Romanoff type theorem
Abstract
Let {ak}k≥1 be an infinite strictly increasing sequence of positive integers with ak| ak+1 for every integer k≥1. Dai and Pan proved that the set of integers of the form p2+q2+ak+al, where p,q are primes and k,l≥1, has positive lower asymptotic density if limsupk→∞(log ak)/k<∞ and supk≥1∑d| ak1/d<∞. They also asked whether the latter condition is necessary. In this paper, we show that this condition is necessary by proving that if supk≥1∑d| ak1/d=∞, then this set has asymptotic density zero.
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BibTeX
@misc{yan2026necessary,
title = {{A necessary condition for a Romanoff type theorem}},
author = {Xiao-Hui Yan and Ya-Ting Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28237},
eprint = {2610.28237},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28237},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A necessary condition for a Romanoff type theorem
AU - Xiao-Hui Yan
AU - Ya-Ting Xu
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28237
UR - https://arxiv.org/abs/2610.28237
AB - Let {a_k}_k≥1 be an infinite strictly increasing sequence of positive integers with a_k| a_k+1 for every integer k≥1. Dai and Pan proved that the set of integers of the form p^2+q^2+a_k+a_l, where p,q are primes and k,l≥1, has positive lower asymptotic density if limsup_k→∞(log a_k)/k<∞ and sup_k≥1∑_d| a_k1/d<∞. They also asked whether the latter condition is necessary. In this paper, we show that this condition is necessary by proving that if sup_k≥1∑_d| a_k1/d=∞, then this set has asymptotic density zero.
ER -