{"paper":{"title":"$\\mathscr{B}$-free sets and dynamics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Aurelia Bartnicka, Joanna Ku{\\l}aga-Przymus, Mariusz Lema\\'nczyk, Stanis{\\l}aw Kasjan","submitted_at":"2015-09-26T16:50:16Z","abstract_excerpt":"Let $B\\subset \\mathbb{N}$ and let $\\eta\\in \\{0,1\\}^\\mathbb{Z}$ be the characteristic function of the set $F_B:=\\mathbb{Z}\\setminus\\bigcup_{b}b\\mathbb{Z}$ of B-free numbers. Consider $(S,X_\\eta)$, where $X_\\eta$ is the closure of the orbit of $\\eta$ under the left shift S. When $B=\\{p^2 : p\\in P\\}$, $(S,X_\\eta)$ was studied by Sarnak. This case + some generalizations, including the case (*) of B infinite, coprime with $\\sum_{b}1/b<\\infty$, were discussed by several authors. For general B, contrary to (*), we may have $X_\\eta\\subsetneq X_B:=\\{x\\in \\{0,1\\}^\\mathbb{Z} : |\\text{supp }x\\bmod b|\\leq "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.08010","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"}