A necessary condition for a Romanoff type theorem

Source abstract

Let $\{a_k\}_{k\ge1}$ be an infinite strictly increasing sequence of positive integers with $a_k\mid a_{k+1}$ for every integer $k\ge1$. 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 $k,l\ge1$, has positive lower asymptotic density if $\limsup_{k\to\infty}\frac{\log a_k}{k}<\infty$ and $\sup_{k\ge1}\sum_{d\mid a_k}\frac1d<\infty$. They also asked whether the latter condition is necessary. In this paper, we show that this condition is necessary by proving that if $\sup_{k\ge1}\sum_{d\mid a_k}\frac1d=\infty$, then this set has asymptotic density zero.