In this paper, we prove an extension of Mahler's theorem, a celebrated result of $p$-adic analysis. Mahler's original result states that a function from $N$ to $Z$ is uniformly continuous for the $p$-adic metric $d_p$ if and only if it can be uniformly approximated by polynomial functions. We prove the same result for functions from $A^*$ to $Z$, where $d_p$ is now the profinite metric defined by $p$-groups (pro-$p$ metric).
@InProceedings{pin_et_al:LIPIcs.STACS.2008.1321, author = {Pin, Jean-Eric and Silva, Pedro V.}, title = {{A Mahler's theorem for functions from words to integers}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {585--596}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1321}, URN = {urn:nbn:de:0030-drops-13212}, doi = {10.4230/LIPIcs.STACS.2008.1321}, annote = {Keywords: \$p\$-adic topology, binomial coefficients, Mahler's theorem, \$p\$-group languages} }
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