A counterexample of two Romanov type conjectures (Q2080965)

It has been shown in 1934 by \textit{N. P. Romanov} [Math. Ann. 109, 668--678 (1934; Zbl 0009.00801)] that the set of numbers of the form \(n=p+2^k\) with prime \(p\) has a positive lower density, and in \textit{H. Li} and \textit{H. Pan} [Acta Arith. 135, No. 2, 137--142 (2008; Zbl 1229.11130)] proved that the same assertion holds for the set \(\{n = 2^p +q \}\) with \(q\) being a prime or a product of two primes. At the end of the last paper two conjectures of Yong-Gao Chen were presented, stating that if \(A,B\) are sets of positive integers, \(C=A+B\), and for sufficiently large \(x\) and some \(c>0\) one has \[ A(\log x/\log 2)B(x) > cx, \] (where \(A(t),B(t)\) count the numbers in \(A\) resp. \(B\) in the interval \([1,t]\)), then the set \(C=\{n = 2^a+b:\ a\in A, b\in B\}\) has a positive lower asymptotic density, and if this inequality holds for infinitely many integers \(x\), then \(C\) has a positive upper density. The authors take for \(A\) the set of all natural numbers and construct a set \(B\) so that the pair \(A,B\) forms a counterexample to both conjectures.