{"paper":{"title":"On self-similar measures with absolutely continuous projections and dimension conservation in each direction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Ariel Rapaport","submitted_at":"2018-09-26T11:56:52Z","abstract_excerpt":"Relying on results due to Shmerkin and Solomyak, we show that outside a $0$-dimensional set of parameters, for every planar homogeneous self-similar measure $\\nu$, with strong separation, dense rotations and dimension greater than $1$, there exists $q>1$ such that $\\{P_{z}\\nu\\}_{z\\in S}\\subset L^{q}(\\mathbb{R})$. Here $S$ is the unit circle and $P_{z}w=\\left\\langle z,w\\right\\rangle $ for $w\\in\\mathbb{R}^{2}$. We then study such measures. For instance, we show that $\\nu$ is dimension conserving in each direction and that the map $z\\rightarrow P_{z}\\nu$ is continuous with respect to the weak top"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.09923","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}