{"paper":{"title":"$\\ell^{p}$ improving estimates for multilinear forms motivated by distance graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"ℓ^p improving estimates for distance graph forms depend only on vertex count in many cases","cross_cats":["math.NT"],"primary_cat":"math.CA","authors_text":"Eyvindur Palsson, Jennifer Smucker","submitted_at":"2026-05-12T17:34:40Z","abstract_excerpt":"We undertake a systematic study of the mapping properties of forms based on distance graphs in $\\mathbb{Z}^{d}$ to see how the structure of a graph, $G$, affects the $\\ell^{p}$ improving estimates of the form, $\\Lambda_{G}$, based on $G$. This extends previous work on $\\ell^{p}$ improving properties for the spherical averaging operator, which corresponds to a distance graph of a single distance. We obtain $\\ell^{p}$ improving estimates for the collection of forms based on all graphs with 2, 3, and 4 vertices, as well as chains and simplexes of any size in $\\mathbb{Z}^{d}$. Surprisingly, certai"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We obtain ℓ^p improving estimates for the collection of forms based on all graphs with 2, 3, and 4 vertices, as well as chains and simplexes of any size in Z^d. Surprisingly, certain mapping properties only seem to depend on the number of vertices in the graph, not its structure.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The distance graphs are defined in Z^d with the standard Euclidean distances, and the multilinear forms satisfy the usual translation-invariance and Fourier multiplier properties assumed in prior spherical averaging work.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"ℓ^p improving estimates are derived for multilinear forms from all distance graphs with 2, 3, or 4 vertices and from chains and simplexes of arbitrary size in Z^d, with some bounds depending only on vertex count rather than graph structure.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"ℓ^p improving estimates for distance graph forms depend only on vertex count in many cases","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f6ec68075268ad3fd7e3e45174f1e3625edc910dfd640d82ef8696378b88db15"},"source":{"id":"2605.12439","kind":"arxiv","version":2},"verdict":{"id":"1f2537ed-ec6c-4ef9-a772-55574e284fa6","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-13T02:30:26.655930Z","strongest_claim":"We obtain ℓ^p improving estimates for the collection of forms based on all graphs with 2, 3, and 4 vertices, as well as chains and simplexes of any size in Z^d. Surprisingly, certain mapping properties only seem to depend on the number of vertices in the graph, not its structure.","one_line_summary":"ℓ^p improving estimates are derived for multilinear forms from all distance graphs with 2, 3, or 4 vertices and from chains and simplexes of arbitrary size in Z^d, with some bounds depending only on vertex count rather than graph structure.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The distance graphs are defined in Z^d with the standard Euclidean distances, and the multilinear forms satisfy the usual translation-invariance and Fourier multiplier properties assumed in prior spherical averaging work.","pith_extraction_headline":"ℓ^p improving estimates for distance graph forms depend only on vertex count in many cases"},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.12439/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-26T12:44:27.931904Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-20T13:01:24.889680Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-20T09:27:35.244563Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T22:21:57.839213Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"c4c13e04c8ec1258e047020915de4dfab8a83cd401d0752909c4429f931877c7"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"acc47c7cd1c998a103266864af68bb0bd9b97c7762439767d48356b89705f905"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}