{"paper":{"title":"A multi-parameter variant of the Erd\\H{o}s distance problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alex Iosevich, Jonathan Passant, Maria Janczak","submitted_at":"2017-12-11T22:49:07Z","abstract_excerpt":"We study the following variant of the Erd\\H{o}s distance problem. Given $E$ and $F$ a point sets in $\\mathbb{R}^d$ and $p = (p_1, \\ldots, p_q)$ with $p_1+ \\cdots + p_q = d$ is an increasing partition of $d$ define $$ B_p(E,F)=\\{(|x_1-y_1|, \\ldots, |x_q-y_q|): x \\in E, y \\in F \\},$$ where $x=(x_1, \\ldots, x_q)$ with $x_i$ in $\\mathbb{R}^{p_i}$. For $p_1 \\geq 2$ it is not difficult to construct $E$ and $F$ such that $|B_{p}(E,F)|=1$. On the other hand, it is easy to see that if $\\gamma_q$ is the best know exponent for the distance problem in $\\mathbb{R}^{p_i}$ that $|B_p(E,E)| \\geq C{|E|}^{\\frac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.04060","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"}