{"paper":{"title":"Controlling Lipschitz functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO","math.MG"],"primary_cat":"math.FA","authors_text":"Andrey Kupavskii, Gabor Tardos, Janos Pach","submitted_at":"2017-04-10T21:49:16Z","abstract_excerpt":"Given any positive integers $m$ and $d$, we say the a sequence of points $(x_i)_{i\\in I}$ in $\\mathbb R^m$ is {\\em Lipschitz-$d$-controlling} if one can select suitable values $y_i\\; (i\\in I)$ such that for every Lipschitz function $f:\\mathbb R^m\\rightarrow \\mathbb R^d$ there exists $i$ with $|f(x_i)-y_i|<1$. We conjecture that for every $m\\le d$, a sequence $(x_i)_{i\\in I}\\subset\\mathbb R^m$ is $d$-controlling if and only if $$\\sup_{n\\in\\mathbb N}\\frac{|\\{i\\in I\\, :\\, |x_i|\\le n\\}|}{n^d}=\\infty.$$ We prove that this condition is necessary and a slightly stronger one is already sufficient for "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.03062","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":""},"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"}