{"paper":{"title":"Diophantine approximation by primes and Landau--Siegel zeros","arxiv_id":"2609.30207","primary_cat":"math.NT","submitted_at":"2026-09-24T17:41:46+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"Let $\\alpha>0$ be an irrational number of finite type and let $\\beta\\in\\mathbb{R}$.","hedge":{"p":0.67,"label":"unclear"},"own_p":0.06,"main_claim_p":0.02,"quantitative":true,"quantitative_p":0.64,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":0.96,"id":"s1","own":true,"role":"result","text":"Assuming the infinitude of Siegel zeros (respectively, sufficiently strong Siegel zeros), we show that for every sufficiently small $\\varepsilon>0$, there exist infinitely many primes $p$ such that \\begin{gather*} \\|\\alpha p+\\beta\\|\\leq p^{-1/3+\\varepsilon} \\qquad\\text{(respectively, }\\|\\alpha p+\\beta\\|\\leq p^{-1/3-1/20}\\text{)}. \\end{gather*}","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.95,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.96,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.04,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":1.0,"text":"Assuming the infinitude of Siegel zeros (respectively, sufficiently strong Siegel zeros), we show that for every sufficiently small $\\varepsilon>0$, there exist infinitely many primes $p$ such that \\begin{gather*} \\|\\alpha p+\\beta\\|\\leq p^{-1/3+\\varepsilon} \\qquad\\text{(respectively, }\\|\\alpha p+\\beta\\|\\leq p^{-1/3-1/20}\\text{)}. \\end{gather*}","location":"abstract s1","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per paper","requests":1,"input_tokens":2456,"cost_usd_reported":0.0},"compiled_at":"2026-09-25","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.03,"states_limitation":0.74,"human_participants":0.01,"controlled_comparison":0.02,"randomized_assignment":0.01,"states_quantitative_result":0.97,"compares_with_prior_results":0.1,"mentions_code_or_data_availability":0.01},"evidential_basis":{"p":1.0,"value":"proof or derivation","probabilities":{"argument":0.0,"literature":0.0,"proof_or_derivation":1.0,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.0}},"contribution_type":{"p":1.0,"value":"theoretical result","probabilities":{"review":0.0,"method_or_tool":0.0,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":1.0,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":1.0,"text":"Assuming the infinitude of Siegel zeros (respectively, sufficiently strong Siegel zeros), we show that for every sufficiently small $\\varepsilon>0$, there exist infinitely many primes $p$ such that \\begin{gather*} \\|\\alpha p+\\beta\\|\\leq p^{-1/3+\\varepsilon} \\qquad\\text{(respectively, }\\|\\alpha p+\\beta\\|\\leq p^{-1/3-1/20}\\text{)}. \\end{gather*}","location":"abstract s1"}},"coverage":{"tier":"abstract","levels_run":["P abstract"],"not_examined":["full text","figures and tables","references and citing sentences","relations to other papers"],"sources_read":["arXiv listing abstract v1"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":2,"sentences_in_abstract":2},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; the figure is how often the published label matches their consensus","sentence_role_agreement":0.92,"main_claim_agreement":0.908,"typed_citation_agreement":0.933,"typed_citation_agreement_95_interval":[0.917,0.945],"review_comments_accurate":0.758,"review_comments_accurate_95_interval":[0.684,0.82],"review_caught_admitted_error_found_or_related":0.448}}