{"paper":{"title":"On iterated product sets with shifts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CO"],"primary_cat":"math.NT","authors_text":"Brandon Hanson, Dmitrii Zhelezov, Oliver Roche-Newton","submitted_at":"2018-01-24T13:39:27Z","abstract_excerpt":"We prove that, for any finite set $A \\subset \\mathbb Q$ with $|AA| \\leq K|A|$ and any positive integer $k$, the $k$-fold product set of the shift $A+1$ satisfies the bound $$| \\{(a_1+1)(a_2+1) \\cdots (a_k+1) : a_i \\in A \\}| \\geq \\frac{|A|^k}{(8k^4)^{kK}}. $$ This result is essentially optimal when $K$ is of the order $c\\log|A|$, for a sufficiently small constant $c=c(k)$.\n  Our main tool is a multiplicative variant of the $\\Lambda$-constants used in harmonic analysis, applied to Dirichlet polynomials."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.07982","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"}