Sumset of three arithmetic progressions in the complex plane (Q6115839)

Let \(\alpha,\beta\) be complex numbers and consider the set \[ S(\alpha,\beta)=\mathbb Z+\mathbb Z\alpha+\mathbb Z\beta=\{a+b\alpha+c\beta|a,b,c\in\mathbb Z\}\] \textit{S. V. Konyagin} et al. [Math. Res. Lett. 30, No. 2, 509--540 (2023; Zbl 07738728)] showed that \(S(\alpha,\beta)\) is everywhere dense in \(\mathbb C\) if \(1,\alpha,\beta\) are linearly independent over \(\mathbb Q\) and \[\mathbb Q(\alpha,\beta)\cap\mathbb R=\mathbb Q.\] The author of the present paper shows that the set \(S(\alpha, \beta)\) is everywhere dense in \(\mathbb C\) if and only if the imaginary parts \(\Im(\alpha),\Im(\beta),\Im(\overline{\alpha}\beta)\) are linearly independent over \(\mathbb Q\). In a symmetric form, for \(\alpha,\beta,\gamma\in\mathbb C\) the set \(\mathbb Z\alpha+\mathbb Z\beta+\mathbb Z\gamma\) is everywhere dense in \(\mathbb C\) if and only if the imaginary parts \(\Im(\alpha\overline{\beta}),\Im(\beta\overline{\gamma}),\Im(\gamma\overline{\alpha})\) are linearly independent over \(\mathbb Q\). For the proof Kronecker's approximation theorem is used.