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We conjecture that for every $m\\le d$, a sequence $(x_i)_{i\\in I}\\subset\\mathbb R^m$ is $d$-controlling if and only if $$\\sup_{n\\in\\mathbb N}\\frac{|\\{i\\in I\\, :\\, |x_i|\\le n\\}|}{n^d}=\\infty.$$ We prove that this condition is necessary and a slightly stronger one is already sufficient for "},"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":"1704.03062","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-04-10T21:49:16Z","cross_cats_sorted":["cs.DM","math.CO","math.MG"],"title_canon_sha256":"7a51b9dce3142160e9b0d562c8c68da955b916485f35ef5466f184eb60dd58df","abstract_canon_sha256":"0b8caf400ed5f71db74126dc73668885de512cee5183df0b55daa074d532e651"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:46.501276Z","signature_b64":"ZKKvdH7F3SM/9MVkB5rUR1GHWtiFrkKmizfM9pxVFWcnyFqpmbsYT1k/dYjiqLK1jAgDPko7QmcjWpQlgUeoAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"76f8329e9541fd9f52ba51ec59d5457926801f70a3ce5329965cd71ebc18b340","last_reissued_at":"2026-05-18T00:08:46.500705Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:46.500705Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Controlling Lipschitz functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO","math.MG"],"primary_cat":"math.FA","authors_text":"Andrey Kupavskii, Gabor Tardos, Janos Pach","submitted_at":"2017-04-10T21:49:16Z","abstract_excerpt":"Given any positive integers $m$ and $d$, we say the a sequence of points $(x_i)_{i\\in I}$ in $\\mathbb R^m$ is {\\em Lipschitz-$d$-controlling} if one can select suitable values $y_i\\; (i\\in I)$ such that for every Lipschitz function $f:\\mathbb R^m\\rightarrow \\mathbb R^d$ there exists $i$ with $|f(x_i)-y_i|<1$. We conjecture that for every $m\\le d$, a sequence $(x_i)_{i\\in I}\\subset\\mathbb R^m$ is $d$-controlling if and only if $$\\sup_{n\\in\\mathbb N}\\frac{|\\{i\\in I\\, :\\, |x_i|\\le n\\}|}{n^d}=\\infty.$$ We prove that this condition is necessary and a slightly stronger one is already sufficient for "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.03062","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1704.03062","created_at":"2026-05-18T00:08:46.500783+00:00"},{"alias_kind":"arxiv_version","alias_value":"1704.03062v2","created_at":"2026-05-18T00:08:46.500783+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.03062","created_at":"2026-05-18T00:08:46.500783+00:00"},{"alias_kind":"pith_short_12","alias_value":"O34DFHUVIH6Z","created_at":"2026-05-18T12:31:34.259226+00:00"},{"alias_kind":"pith_short_16","alias_value":"O34DFHUVIH6Z6UV2","created_at":"2026-05-18T12:31:34.259226+00:00"},{"alias_kind":"pith_short_8","alias_value":"O34DFHUV","created_at":"2026-05-18T12:31:34.259226+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/O34DFHUVIH6Z6UV2KHWFTVKFPE","json":"https://pith.science/pith/O34DFHUVIH6Z6UV2KHWFTVKFPE.json","graph_json":"https://pith.science/api/pith-number/O34DFHUVIH6Z6UV2KHWFTVKFPE/graph.json","events_json":"https://pith.science/api/pith-number/O34DFHUVIH6Z6UV2KHWFTVKFPE/events.json","paper":"https://pith.science/paper/O34DFHUV"},"agent_actions":{"view_html":"https://pith.science/pith/O34DFHUVIH6Z6UV2KHWFTVKFPE","download_json":"https://pith.science/pith/O34DFHUVIH6Z6UV2KHWFTVKFPE.json","view_paper":"https://pith.science/paper/O34DFHUV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1704.03062&json=true","fetch_graph":"https://pith.science/api/pith-number/O34DFHUVIH6Z6UV2KHWFTVKFPE/graph.json","fetch_events":"https://pith.science/api/pith-number/O34DFHUVIH6Z6UV2KHWFTVKFPE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/O34DFHUVIH6Z6UV2KHWFTVKFPE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/O34DFHUVIH6Z6UV2KHWFTVKFPE/action/storage_attestation","attest_author":"https://pith.science/pith/O34DFHUVIH6Z6UV2KHWFTVKFPE/action/author_attestation","sign_citation":"https://pith.science/pith/O34DFHUVIH6Z6UV2KHWFTVKFPE/action/citation_signature","submit_replication":"https://pith.science/pith/O34DFHUVIH6Z6UV2KHWFTVKFPE/action/replication_record"}},"created_at":"2026-05-18T00:08:46.500783+00:00","updated_at":"2026-05-18T00:08:46.500783+00:00"}