{"paper":{"title":"Narrow progressions in the primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tamar Ziegler, Terence Tao","submitted_at":"2014-09-04T06:07:51Z","abstract_excerpt":"In a previous paper of the authors, we showed that for any polynomials $P_1,\\dots,P_k \\in \\Z[\\mathbf{m}]$ with $P_1(0)=\\dots=P_k(0)$ and any subset $A$ of the primes in $[N] = \\{1,\\dots,N\\}$ of relative density at least $\\delta>0$, one can find a \"polynomial progression\" $a+P_1(r),\\dots,a+P_k(r)$ in $A$ with $0 < |r| \\leq N^{o(1)}$, if $N$ is sufficiently large depending on $k,P_1,\\dots,P_k$ and $\\delta$. In this paper we shorten the size of this progression to $0 < |r| \\leq \\log^L N$, where $L$ depends on $k,P_1,\\dots,P_k$ and $\\delta$. In the linear case $P_i = (i-1)\\mathbf{m}$, we can take "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.1327","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}