{"paper":{"title":"Polynomial Szemer\\'edi for sets with large Hausdorff dimension on the Torus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Guo-Dong Hong","submitted_at":"2025-07-18T23:31:10Z","abstract_excerpt":"Let $\\mathbb{P}= \\{P_1, \\cdots, P_{k}\\in \\mathbb{R}[y]\\}$ be a collection of polynomials with distinct degrees and zero constant terms. We proved that there exists $\\epsilon=\\epsilon(\\mathbb{P})>0$ such that, for any compact set $E \\subset \\mathbb{T}$ with dim(E)$>1-\\epsilon$, we can find $y\\neq 0$ so that $\\{x,x+P_1(y), \\cdots,x+P_k(y)\\} \\subset E$. The proof relies on a suitable version of the Sobolev smoothing inequality with ideas adapted from Peluse \\cite{P19}, Durcik and Roos \\cite{DR24}, and Krause, Mirek, Peluse, and Wright \\cite{KMPW24}. As a byproduct of our Sobolev smoothing inequal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.14407","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/2507.14407/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"}