Additive Structure of Sparse-Digit Fractal Sets: Sumsets, Difference Sets, and Dimension Jumps

We study the additive and fractal structure of digit-restricted subsets of the unit interval, \( A_D=\Bigl\{ \sum_{n=1}^\infty a_n b^{-n} : a_n\in D\subseteq\{0,\dots,b-1\},\ |D|\ge2\Bigr\} \) defined by allowing only digits from D in base-b expansions. These sparse-digit sets generalize the middle-third Cantor set and include a wide range of missing-digit and structured-digit fractals. We develop a rigorous carry-propagation framework for base-b digit arithmetic, give sharp combinatorial criteria for intervals in \( A_D+A_D \) and \( A_D-A_D \), expand the proof of the similarity-dimension formula, and strengthen the dimension-jump theorem for iterated sumsets \( A_D^{(k)} \) by providing full justification and relevant references. A new Carry Stabilization Lemma, an expanded interval-criterion proof, and a related-work section situate these results within the literature on sumsets of Cantor sets, fractal addition theorems, and digit-based self-similar sets.