{"paper":{"title":"Visibility and directions in quasicrystals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MG","math.MP","math.NT"],"primary_cat":"math.DS","authors_text":"Andreas Str\\\"ombergsson, Jens Marklof","submitted_at":"2014-04-06T11:09:54Z","abstract_excerpt":"It is well known that a positive proportion of all points in a $d$-dimensional lattice is visible from the origin, and that these visible lattice points have constant density in $\\mathbb{R}^d$. In the present paper we prove an analogous result for a large class of quasicrystals, including the vertex set of a Penrose tiling. We furthermore establish that the statistical properties of the directions of visible points are described by certain $\\operatorname{SL}(d,\\mathbb{R})$-invariant point processes. Our results imply in particular existence and continuity of the gap distribution for directions"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.1564","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"}