{"paper":{"title":"Dimension of Bernoulli Convolutions in $\\mathbb{R}^{d}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.DS","authors_text":"Ariel Rapaport, Haojie Ren","submitted_at":"2024-06-08T15:21:15Z","abstract_excerpt":"For $(\\lambda_{1},...,\\lambda_{d})=\\lambda\\in(0,1)^{d}$ with $\\lambda_{1}>...>\\lambda_{d}$, denote by $\\mu_{\\lambda}$ the Bernoulli convolution associated to $\\lambda$. That is, $\\mu_{\\lambda}$ is the distribution of the random vector $\\sum_{n\\ge0}\\pm\\left(\\lambda_{1}^{n},...,\\lambda_{d}^{n}\\right)$, where the $\\pm$ signs are chosen independently and with equal weight. Assuming for each $1\\le j\\le d$ that $\\lambda_{j}$ is not a root of a polynomial with coefficients $\\pm1,0$, we prove that the dimension of $\\mu_{\\lambda}$ equals $\\min\\left\\{ \\dim_{L}\\mu_{\\lambda},d\\right\\} $, where $\\dim_{L}\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.05495","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2406.05495/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}