{"paper":{"title":"A new perspective on the distance problem over prime fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.CO","authors_text":"Alex Iosevich, Doowon Koh, Thang Pham","submitted_at":"2019-05-10T14:06:32Z","abstract_excerpt":"Let $\\mathbb{F}_p$ be a prime field, and ${\\mathcal E}$ a set in $\\mathbb{F}_p^2$. Let $\\Delta({\\mathcal E})=\\{||x-y||: x,y \\in {\\mathcal E} \\}$, the distance set of ${\\mathcal E}$. In this paper, we provide a quantitative connection between the distance set $\\Delta({\\mathcal E})$ and the set of rectangles determined by points in ${\\mathcal E}$. As a consequence, we obtain a new lower bound on the size of $\\Delta({\\mathcal E})$ when ${\\mathcal E}$ is not too large, improving a previous estimate due to Lund and Petridis and establishing an approach that should lead to significant further improv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.04179","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"}