{"paper":{"title":"Tinkering with Lattices: A New Take on the Erd\\H{o}s Distance Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Elena Kim, Elzbieta Boldyriew, Eyvindur Palsson, Fernando Trejos Su\\'arez, Jason Zhao, Sean Sovine, Steven J. Miller","submitted_at":"2020-09-25T22:04:09Z","abstract_excerpt":"The Erd\\H{o}s distance problem concerns the least number of distinct distances that can be determined by $N$ points in the plane. The integer lattice with $N$ points is known as \\textit{near-optimal}, as it spans $\\Theta(N/\\sqrt{\\log(N)})$ distinct distances, the lower bound for a set of $N$ points (Erd\\H{o}s, 1946). The only previous non-asymptotic work related to the Erd\\H{o}s distance problem that has been done was for $N \\leq 13$. We take a new non-asymptotic approach to this problem in a model case, studying the distance distribution, or in other words, the plot of frequencies of each dis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.12450","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2009.12450/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}