{"paper":{"title":"Invariant random compacts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Continuous actions on compact metric spaces can be IC-rigid, forcing every invariant random compact to be almost surely finite or the entire space.","cross_cats":[],"primary_cat":"math.DS","authors_text":"Bryna Kra, Scott Schmieding","submitted_at":"2026-05-05T17:13:37Z","abstract_excerpt":"For a compact metric space $X$ with a group $G$ acting on it continuously, an invariant random compact is a Borel probability measure on the space of nonempty compact subsets of $X$ that is invariant under the action of $G$. The action is IC-rigid if, with respect to every invariant random compact, every compact set is almost surely either finite or $X$. We give sufficient conditions for an action to be IC-rigid, and show there are natural examples of such actions. We further consider a notion of weak IC-rigidity, and prove that the Chacon system is weakly IC-rigid but not IC-rigid. As an appl"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We give sufficient conditions for an action to be IC-rigid, and show there are natural examples of such actions. We further consider a notion of weak IC-rigidity, and prove that the Chacon system is weakly IC-rigid but not IC-rigid. As an application, we prove results concerning multiplicative largeness of dilations of sets on the circle.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The setup assumes X is a compact metric space and the G-action is continuous, which permits the space of nonempty compact subsets to carry a natural topology and invariant measures; additionally, the specific dynamical properties of the Chacon system must distinguish weak from full rigidity.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Group actions on compact metric spaces are IC-rigid under stated conditions, with the Chacon system weakly IC-rigid, yielding results on multiplicative largeness of dilated sets.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Continuous actions on compact metric spaces can be IC-rigid, forcing every invariant random compact to be almost surely finite or the entire space.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"2eb505f1570d5f78be1ba3d732f0d75b7ae33ee9004d85a825711644cd2fa952"},"source":{"id":"2605.03993","kind":"arxiv","version":2},"verdict":{"id":"a27724db-c5aa-44ed-9a71-9d1f93d7d2a7","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T04:08:40.016980Z","strongest_claim":"We give sufficient conditions for an action to be IC-rigid, and show there are natural examples of such actions. We further consider a notion of weak IC-rigidity, and prove that the Chacon system is weakly IC-rigid but not IC-rigid. As an application, we prove results concerning multiplicative largeness of dilations of sets on the circle.","one_line_summary":"Group actions on compact metric spaces are IC-rigid under stated conditions, with the Chacon system weakly IC-rigid, yielding results on multiplicative largeness of dilated sets.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The setup assumes X is a compact metric space and the G-action is continuous, which permits the space of nonempty compact subsets to carry a natural topology and invariant measures; additionally, the specific dynamical properties of the Chacon system must distinguish weak from full rigidity.","pith_extraction_headline":"Continuous actions on compact metric spaces can be IC-rigid, forcing every invariant random compact to be almost surely finite or the entire space."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.03993/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T12:40:11.587869Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-20T00:01:21.197560Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:51:18.326079Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"bc403b431acde38538de72617503851134582416c246a7cfbb422d5c96e4d895"},"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"}