PreprintCC BY 4.0arXiv
Optimal local convergence criteria for integer and Gaussian integer continued fractions
Abstract
The objective of this work is to determine optimal local restrictions on the coefficients of integer and Gaussian integer continued fractions that imply convergence. We identify all minimal restrictions involving words of length two in the integer case, and we identify all reversible minimal restrictions of length two in the Gaussian integer case. In the integer setting, our classification is equivalent to a classification of minimal unavoidable words of length two in Conway--Coxeter quiddity sequences. We also construct a canonical set of restrictions of infinite cardinality that is strictly stronger than every finite set of restrictions.
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BibTeX
@misc{short2026optimal,
title = {{Optimal local convergence criteria for integer and Gaussian integer continued fractions}},
author = {Ian Short and Margaret Stanier and Matty van Son and Andrei Zabolotskii},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.13199},
eprint = {2608.13199},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.13199},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal local convergence criteria for integer and Gaussian integer continued fractions
AU - Ian Short
AU - Margaret Stanier
AU - Matty van Son
AU - Andrei Zabolotskii
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2608.13199
UR - https://arxiv.org/abs/2608.13199
AB - The objective of this work is to determine optimal local restrictions on the coefficients of integer and Gaussian integer continued fractions that imply convergence. We identify all minimal restrictions involving words of length two in the integer case, and we identify all reversible minimal restrictions of length two in the Gaussian integer case. In the integer setting, our classification is equivalent to a classification of minimal unavoidable words of length two in Conway--Coxeter quiddity sequences. We also construct a canonical set of restrictions of infinite cardinality that is strictly stronger than every finite set of restrictions.
ER -