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 where AD=∑n=1∞anb−n:an∈D⊆{0,…,b−1},|D|≥2, defined by allowing only digits from D in base-b expansions. These sets generalize the middle-third Cantor set and include a wide range of missing-digit and structured-digit fractals. We develop a rigorous framework for base-b digit arithmetic that separates purely discrete digit-combinatorics from carry effects. We give sharp sufficient criteria for intervals in AD+AD and AD−AD via carry-free digit blocks, establish an arithmetic obstruction to interval formation for all iterated sumsets, and prove a dimension-jump dichotomy: either a gcd obstruction prevents intervals for every k, or else some iterated sumset AD(k) contains an interval, and hence has full Hausdorff dimension 1. We also discuss the similarity-dimension formula under the open set condition, include definitions and preliminaries for a broad audience, and situate the results within classical and modern literature on Cantor sets and sumsets of self-similar sets.