A generalization of the Romanoff theorem
Yuchen Ding, Wenguang Zhai · International Journal of Number Theory · 2025
Let [Formula: see text] be the set of primes and [Formula: see text] the set of positive integers. Let also [Formula: see text] be positive real numbers and [Formula: see text] the set of odd integers which can be represented as [Formula: see text] where [Formula: see text] and [Formula: see text]. Recently, Chen and Xu proved that the set [Formula: see text] has positive lower asymptotic density, provided that [Formula: see text] and at least one of [Formula: see text] is an integer. Their result reduces to the famous theorem of Romanoff by taking [Formula: see text] In this paper, we remove the unnecessary condition: ‘at least one of [Formula: see text] is an integer’.