On the densities of some sequences
In 1934, \textit{N. P. Romanoff} [Math. Ann. 109, 668-678 (1934; Zbl 0009.00801)] proved that the integers of the form \(p+2^ m\), with \(p\) prime, have positive density in \(\mathbb{Z}\). In this paper, the authors generalize this result to Gaussian integers: If \(\Lambda_ 1\) and \(\Lambda_ 2\) are fixed Gaussian integers with \(\Lambda^ 4_ 2=1\) and \(\left| \Lambda_ 1-\sqrt{\Lambda^ 2_ 1+4\Lambda_ 2}\right|\neq\left|\Lambda_ 1+\sqrt{\Lambda^ 2_ 1 +4\Lambda_ 2}\right|\) then \(\{\pi+\varphi_ n\}\) has positive density in \(\mathbb{Z}[i]\), where \(\pi\) varies over the Gaussian primes and \(\varphi_ n\) is defined by \(\varphi_ 0=0\), \(\varphi_ 1=1\), \(\varphi_{n+2}=\Lambda_ 1\varphi_{n+1}+\Lambda_ 2\varphi_ n\). Related questions are also discussed.
scientific article; zbMATH DE number 166049