{"ID":2872582,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.08532","arxiv_id":"2509.08532","title":"Stabilisability and beta representations","abstract":"We consider a one dimensional affine switched system obtained from a formal limit of a two dimensional linear system. We show this is equivalent to minimising the average digit in beta representations with unrestricted digits. We give a countable set of $β$ for which the result is given by the usual (greedy) beta expansion, an interval of values for which it is strictly less, and a conditional lower bound for all $β$.","short_abstract":"We consider a one dimensional affine switched system obtained from a formal limit of a two dimensional linear system. We show this is equivalent to minimising the average digit in beta representations with unrestricted digits. We give a countable set of $β$ for which the result is given by the usual (greedy) beta expan...","url_abs":"https://arxiv.org/abs/2509.08532","url_pdf":"https://arxiv.org/pdf/2509.08532v1","authors":"[\"Carl P. Dettmann\"]","published":"2025-09-10T12:19:45Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.DS\"]","methods":"[]","has_code":false}