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IC-rigidity and generalizations of a theorem of Mahler
Abstract
Mahler showed that for every irrational α and every base b≥ 2, there is a single positive integer n such that every digit in {0, …, b−1} occurs infinitely often in the base-b expansion of nα. Furstenberg later observed that dynamical methods can be used to give a different proof. Building on this perspective, we give generalizations of Mahler's theorem in two directions: the multipliers are a syndetic subset of certain multiplicative semigroups, and analogous versions hold in higher dimensions. Our main tool is IC-rigidity, a type of rigidity which is a hybrid of topological and measurable constraints.
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BibTeX
@misc{kra2026icrigidity,
title = {{IC-rigidity and generalizations of a theorem of Mahler}},
author = {Bryna Kra and Scott Schmieding},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05607},
eprint = {2611.05607},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05607},
}
RIS
TY - UNPB
TI - IC-rigidity and generalizations of a theorem of Mahler
AU - Bryna Kra
AU - Scott Schmieding
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05607
UR - https://arxiv.org/abs/2611.05607
AB - Mahler showed that for every irrational α and every base b≥ 2, there is a single positive integer n such that every digit in {0, …, b−1} occurs infinitely often in the base-b expansion of nα. Furstenberg later observed that dynamical methods can be used to give a different proof. Building on this perspective, we give generalizations of Mahler's theorem in two directions: the multipliers are a syndetic subset of certain multiplicative semigroups, and analogous versions hold in higher dimensions. Our main tool is IC-rigidity, a type of rigidity which is a hybrid of topological and measurable constraints.
ER -