{"paper":{"title":"Dimensions of sets which uniformly avoid arithmetic progressions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.CA","authors_text":"Han Yu, Jonathan M. Fraser, Kota Saito","submitted_at":"2017-05-09T13:56:04Z","abstract_excerpt":"We provide estimates for the dimensions of sets in $\\mathbb{R}$ which uniformly avoid finite arithmetic progressions. More precisely, we say $F$ uniformly avoids arithmetic progressions of length $k \\geq 3$ if there is an $\\epsilon>0$ such that one cannot find an arithmetic progression of length $k$ and gap length $\\Delta>0$ inside the $\\epsilon \\Delta$ neighbourhood of $F$. Our main result is an explicit upper bound for the Assouad (and thus Hausdorff) dimension of such sets in terms of $k$ and $\\epsilon$. In the other direction, we provide examples of sets which uniformly avoid arithmetic pr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.03335","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/1705.03335/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"}