{"paper":{"title":"Interior of sums of planar sets and curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"K\\'aroly Simon, Krystal Taylor","submitted_at":"2017-07-05T14:34:55Z","abstract_excerpt":"Recently, considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the interior $\\left(A+\\Gamma\\right)^{\\circ}$, when $\\Gamma$ is a piecewise $\\mathcal{C}^2$ curve and $A\\subset \\mathbb{R}^2.$ To begin, we give an example of a very large (full-measure, dense, $G_\\delta$) set $A$ such that $\\left(A+S^1\\right)^{\\circ}=\\emptyset$, where $S^1$ denotes the unit circle. This suggests that merely the size of $A$ does not guarantee that $(A+S^1)^{\\circ }\\ne\\emptyset$. If, however, we assume that $A$ is a kind of generalized product of two r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.01420","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"}