Times two, three, five orbits on $\mathbb{T}^2$

In this paper, we study orbit closures under diagonal torus actions. We show that if $(x,y)\in\mathbb{T}^2$ is not contained in any rational lines, then its orbit under the $\times 2, \times 3, \times 5$ actions is dense in $\mathbb{T}^2.$

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