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We study sets of local dimensions for self-similar measures in $\mathbb{R}$satisfying the finite neighbour condition, which is formally stronger than theweak separation condition but satisfied in all known examples. Under a mildtechnical assumption, we establish that the set of attainable local dimensionsis a finite union of (possibly singleton) compact intervals. The number ofintervals is bounded above by the number of non-trivial maximal stronglyconnected components of a finite directed graph construction depending only onthe governing iterated function system. We also explain how our results allowcomputations of the sets of local dimensions in many explicit cases. Thiscontextualizes and generalizes a vast amount of prior work on sets of localdimensions for self-similar measures satisfying the weak separation condition.

Local Dimensions of Self-similar Measures Satisfying the Finite Neighbour Condition