{"paper":{"title":"On additive shifts of multiplicative almost-subgroups in finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dmitrii Zhelezov","submitted_at":"2015-07-20T16:07:15Z","abstract_excerpt":"We prove that for sets $A, B, C \\subset \\mathbb{F}_p$ with $|A|=|B|=|C| \\leq \\sqrt{p}$ and a fixed $0 \\neq d \\in \\mathbb{F}_p$ holds\n  $$\n  \\max(|AB|, |(A+d)C|) \\gg|A|^{1+1/26}.\n  $$\n  In particular,\n  $$\n  |A(A+1)| \\gg |A|^{1 + 1/26}\n  $$\n  and\n  $$\n  \\max(|AA|, |(A+1)(A+1)|) \\gg |A|^{1 + 1/26}.\n  $$\n  The first estimate improves the bound by Roche-Newton and Jones.\n  In the general case of a field of order $q = p^m$ we obtain similar estimates with the exponent $1+1/559 + o(1)$ under the condition that $AB$ does not have large intersection with any subfield coset, answering a question of Shp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.05548","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":""},"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"}