In this note, we give a necessary and sufficient condition on α, β, γ ∈ ℂ under which the sumset of three infinite arithmetic progressions ℤα+ℤβ+ℤγ is everywhere dense in ℂ. This is the case if and only if the imaginary parts \(\mathfrak{I}\left(\alpha \overline{\beta }\right),\mathfrak{I}\left(\beta \overline{\gamma }\right),\mathfrak{I}\left(\gamma \overline{\alpha }\right)\) are linearly independent over ℚ. This completes a recent result of Konyagin, Sha, Shparlinski, and Stewart.

中文翻译：

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复平面中三个等差级数的总和

在这篇笔记中，我们给出了α, β, γ ∈ ℂ的充分必要条件，在此条件下，三个无限等差数列的和集合 ℤ α +ℤ β +ℤ γ在 ℂ 中处处稠密。当且仅当虚部\(\mathfrak{I}\left(\alpha \overline{\beta }\right),\mathfrak{I}\left(\beta \overline{\gamma }\ right),\mathfrak{I}\left(\gamma \overline{\alpha }\right)\)在 ℚ 上线性无关。这完成了 Konyagin、Sha、Shparlinski 和 Stewart 最近的一个结果。