Let \(\mathcal{C} + \mathcal{C}\) denote the sumset of the Cantor set \(\mathcal{C}\). That is, \[\mathcal{C} + \mathcal{C} = \{x+y \, : \, x, y \in \mathcal{C}\}.\] Which of the following statements is true about \(\mathcal{C} + \mathcal{C}\)?

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