Existence of density

Question

Suppose $ \{ D_i : i \geq 1 \} $ be a sequence of i.i.d. random variables taking values $ \{ 0, 1, 2, \dotsc, K-1 \} $ where $ K \geq 3 $ is a positive integer with probabilities $ \mathbb{P} ( D_1 = j) = \alpha_j $ for $ j = 0, 1, \dotsc, K-1$. Consider the random variable \begin{equation*} X = \sum_{ i = 1}^{\infty} D_i K^{-i} . \end{equation*} Under what conditions on $ \alpha_j$'s, $ X $ will admit a density i.e., its distribution will be absolutely continuous with respect to Lebesgue measure?