{"paper":{"title":"Decomposing with smooth sets","license":"","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Juris Stepr\\=ans","submitted_at":"1995-01-07T00:00:00Z","abstract_excerpt":"A subset of Euclidean space will be said to be $n$-smooth if it has an $n$-dimensional tangent plane at each of its points. Let ${\\frak d}_n$ denote the least number $n$-smooth sets into which $n+1$-dimensional Euclidean space can be decomposed. For each $n$ it is shown to be consistent that ${\\frak d}_n > {\\frak d}_{n+1} $.  Moreover, the inequalities ${\\frak d}_{n+1}^+ \\geq ${\\frak d}_n$ are established where ${\\frak d}_1$ is defined to be the continuum. The cardinal invariant ${\\frak d}_2$ is shown to be the same as the least $\\kappa$ such that each continuous function from the reals to the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9501204","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"}