{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:TG3DTUVCD6R6PZ6A3Y5IH3KOVR","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"80c3db6c2064a4bc661eb31cb57ba5c9c9869b0eb997fe5130e6366790fc5213","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-09-17T21:42:37Z","title_canon_sha256":"7971808f7cb21f6d54161a73c67ae04417fe100ccd633f5bc9a1a9914ff0d5a7"},"schema_version":"1.0","source":{"id":"0909.3324","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.3324","created_at":"2026-05-18T04:17:03Z"},{"alias_kind":"arxiv_version","alias_value":"0909.3324v3","created_at":"2026-05-18T04:17:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.3324","created_at":"2026-05-18T04:17:03Z"},{"alias_kind":"pith_short_12","alias_value":"TG3DTUVCD6R6","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"TG3DTUVCD6R6PZ6A","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"TG3DTUVC","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:d8c3c1c01b4e700ab7c4edd696aadd432fd6fa6cf3d36c7eae1d4d4c1b7931bf","target":"graph","created_at":"2026-05-18T04:17:03Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $1<q<2$ and \\[ \\Lambda(q)={\\sum_{k=0}^n a_kq^k\\mid a_k\\in\\{-1,0,1\\}, n\\ge1}. \\] It is well known that if $q$ is not a root of a polynomial with coefficients $0,\\pm1$, then $\\Lambda(q)$ is dense in $\\mathbb{R}$. We give several sufficient conditions for the denseness of $\\Lambda(q)$ when $q$ is a root of such a polynomial. In particular, we prove that if $q$ is not a Perron number or it has a conjugate $\\alpha$ such that $q|\\alpha|<1$, then $\\Lambda(q)$ is dense in $\\mathbb{R}$.","authors_text":"Boris Solomyak, Nikita Sidorov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-09-17T21:42:37Z","title":"On the topology of sums in powers of an algebraic number"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.3324","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:e348be5e75f789e05c2f505778d104ad8c8c71eaecf0cc15005efc98586f642e","target":"record","created_at":"2026-05-18T04:17:03Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"80c3db6c2064a4bc661eb31cb57ba5c9c9869b0eb997fe5130e6366790fc5213","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-09-17T21:42:37Z","title_canon_sha256":"7971808f7cb21f6d54161a73c67ae04417fe100ccd633f5bc9a1a9914ff0d5a7"},"schema_version":"1.0","source":{"id":"0909.3324","kind":"arxiv","version":3}},"canonical_sha256":"99b639d2a21fa3e7e7c0de3a83ed4eac7038ae39db28878f91f7df66ed2eb49b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"99b639d2a21fa3e7e7c0de3a83ed4eac7038ae39db28878f91f7df66ed2eb49b","first_computed_at":"2026-05-18T04:17:03.479579Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:17:03.479579Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eJrNRkH30ncMFtTVI2JPrhejZdU8d1C+Z7Bwt+dWuja95ToZb/pX2TJlJwlVKfdGiZ7R3TOnpK+4UCRSpRRwCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:17:03.480099Z","signed_message":"canonical_sha256_bytes"},"source_id":"0909.3324","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e348be5e75f789e05c2f505778d104ad8c8c71eaecf0cc15005efc98586f642e","sha256:d8c3c1c01b4e700ab7c4edd696aadd432fd6fa6cf3d36c7eae1d4d4c1b7931bf"],"state_sha256":"dec0a24f8bb8d1aa2e89d7a3bc7d114909e3ce7400c0cfc76f284e1999b95dab"}