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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2406.05495","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2024-06-08T15:21:15Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"e8ed58c0123bc6e569c1afe18694e45c09f0dd32d7bf462304615053b1edeaf3","abstract_canon_sha256":"4f91d14053b87c22aa9d1889b7c710746e509b73346b8cffbab2f8196d3966ea"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:29:06.500496Z","signature_b64":"yq0Rlq9Xa8hSSK+7KueVhTIW5AzWTV6SdEHYp0rimZCN8ALrOxz3J+O+acDY1c9xwhJW28zt3WYjrwxTCqZ+BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"22d786daed79b71ff8cc5d77bff442dda8fddee959f9512e1f27ad5f0e215dc3","last_reissued_at":"2026-07-05T08:29:06.500080Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:29:06.500080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2406.05495","created_at":"2026-07-05T08:29:06.500120+00:00"},{"alias_kind":"arxiv_version","alias_value":"2406.05495v1","created_at":"2026-07-05T08:29:06.500120+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2406.05495","created_at":"2026-07-05T08:29:06.500120+00:00"},{"alias_kind":"pith_short_12","alias_value":"ELLYNWXNPG3R","created_at":"2026-07-05T08:29:06.500120+00:00"},{"alias_kind":"pith_short_16","alias_value":"ELLYNWXNPG3R76GM","created_at":"2026-07-05T08:29:06.500120+00:00"},{"alias_kind":"pith_short_8","alias_value":"ELLYNWXN","created_at":"2026-07-05T08:29:06.500120+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2501.17378","citing_title":"Dimension of diagonal self-affine measures with exponentially separated projections","ref_index":49,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ELLYNWXNPG3R76GMLV3375CC3W","json":"https://pith.science/pith/ELLYNWXNPG3R76GMLV3375CC3W.json","graph_json":"https://pith.science/api/pith-number/ELLYNWXNPG3R76GMLV3375CC3W/graph.json","events_json":"https://pith.science/api/pith-number/ELLYNWXNPG3R76GMLV3375CC3W/events.json","paper":"https://pith.science/paper/ELLYNWXN"},"agent_actions":{"view_html":"https://pith.science/pith/ELLYNWXNPG3R76GMLV3375CC3W","download_json":"https://pith.science/pith/ELLYNWXNPG3R76GMLV3375CC3W.json","view_paper":"https://pith.science/paper/ELLYNWXN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2406.05495&json=true","fetch_graph":"https://pith.science/api/pith-number/ELLYNWXNPG3R76GMLV3375CC3W/graph.json","fetch_events":"https://pith.science/api/pith-number/ELLYNWXNPG3R76GMLV3375CC3W/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ELLYNWXNPG3R76GMLV3375CC3W/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ELLYNWXNPG3R76GMLV3375CC3W/action/storage_attestation","attest_author":"https://pith.science/pith/ELLYNWXNPG3R76GMLV3375CC3W/action/author_attestation","sign_citation":"https://pith.science/pith/ELLYNWXNPG3R76GMLV3375CC3W/action/citation_signature","submit_replication":"https://pith.science/pith/ELLYNWXNPG3R76GMLV3375CC3W/action/replication_record"}},"created_at":"2026-07-05T08:29:06.500120+00:00","updated_at":"2026-07-05T08:29:06.500120+00:00"}