{"paper":{"title":"Discrete spheres and arithmetic progressions in product sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dmitrii Zhelezov","submitted_at":"2015-10-19T10:05:40Z","abstract_excerpt":"We prove that if $B$ is a set of $N$ positive integers such that $B\\cdot B$ contains an arithmetic progression of length $M$, then for some absolute $C > 0$,\n  $$\n  \\pi(M) + C \\frac {M^{2/3}}{\\log^2 M} \\leq N,\n  $$ where $\\pi$ is the prime counting function.\n  This improves on previously known bounds of the form $N = \\Omega(\\pi(M))$ and gives a bound which is sharp up to the second order term, as Pach and S\\'andor gave an example for which\n  $$\n  N < \\pi(M)+ O\\left(\\frac {M^{2/3}}{\\log^2 M} \\right).\n  $$\n  The main new tool is a reduction of the original problem to the question of approximate "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.05411","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"}