On the dimension of orthogonal projections of self-similar measures

Let (Formula presented.) be a self-similar measure on (Formula presented.), (Formula presented.), and let (Formula presented.) be an orthogonal projection onto a (Formula presented.) -dimensional subspace. We formulate a criterion on the action of the group generated by the orthogonal parts of the iterated function system on (Formula presented.), and show that it ensures that the dimension of (Formula presented.) is preserved; this significantly refines previous results by Hochman–Shmerkin (2012) and Falconer–Jin (2014), and is sharp for projections to lines and hyperplanes. A key ingredient in the proof is an application of a restricted projection theorem of Gan–Guo–Wang (2024).