{"paper":{"title":"Towards Resolving Keller's Cube Tiling Conjecture in Dimension Seven","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Andrzej P. Kisielewicz","submitted_at":"2017-01-25T04:36:32Z","abstract_excerpt":"A cube tiling of $\\mathbb{R}^d$ is a family of pairwise disjoint cubes $[0,1)^d+T=\\{[0,1)^d+t\\colon t\\in T\\}$ such that $\\bigcup_{t\\in T}([0,1)^d+t)=\\mathbb{R}^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if $|t_j-s_j|=1$ for some $j\\in [d]=\\{1,\\ldots, d\\}$ and $t_i=s_i$ for every $i\\in [d]\\setminus \\{j\\}$. In $1930$, Keller conjectured that in every cube tiling of $\\mathbb{R}^d$ there is a twin pair. Keller's conjecture is true for dimensions $d\\leq 6$ and false for all dimensions $d\\geq 8$. For $d=7$ the conjecture is still open. Let $x\\in \\mathbb{R}^d$, $i\\in [d]$, and let "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.07155","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"}