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MAILLET’S PROPERTY AND MAHLER’S CONJECTURE ON LIOUVILLE NUMBERS FAIL FOR MATRICES

Part of: Diophantine approximation, transcendental number theory Series expansions

Published online by Cambridge University Press:  10 March 2025

JOHANNES SCHLEISCHITZ Open the ORCID record for JOHANNES SCHLEISCHITZ [Opens in a new window]
JOHANNES SCHLEISCHITZ*
Affiliation:
Middle East Technical University, Northern Cyprus Campus, Kalkanli, Güzelyurt, Türkiye e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

In the early 1900s, Maillet [Introduction a la theorie des nombres transcendants et des proprietes arithmetiques des fonctions (Gauthier–Villars, Paris, 1906)] proved that the image of any Liouville number under a rational function with rational coefficients is again a Liouville number. The analogous result for quadratic Liouville matrices in higher dimensions turns out to fail. In fact, using a result by Kleinbock and Margulis [‘Flows on homogeneous spaces and Diophantine approximation on manifolds’, Ann. of Math. (2) 148(1) (1998), 339–360], we show that among analytic matrix functions in dimension $n\ge 2$, Maillet’s invariance property is only true for Möbius transformations with special coefficients. This implies that the analogue in higher dimensions of an open question of Mahler on the existence of transcendental entire functions with Maillet’s property has a negative answer. However, extending a topological argument of Erdős [‘Representations of real numbers as sums and products of Liouville numbers’, Michigan Math. J. 9 (1962), 59–60], we prove that for any injective continuous self-mapping on the space of rectangular matrices, many Liouville matrices are mapped to Liouville matrices. Dropping injectivity, we consider setups similar to Alniaçik and Saias [‘Une remarque sur les $G_{\delta }$-denses’, Arch. Math. (Basel) 62(5) (1994), 425–426], and show that the situation depends on the matrix dimensions $m,n$. Finally, we discuss extensions of a related result by Burger [‘Diophantine inequalities and irrationality measures for certain transcendental numbers’, Indian J. Pure Appl. Math. 32 (2001), 1591–1599] to quadratic matrices. We state several open problems along the way.

Keywords

Liouville numberirrationality exponenttranscendental function

MSC classification

Primary: 30B10: Power series (including lacunary series) 11J13: Simultaneous homogeneous approximation, linear forms
Type
Research Article
Information
Journal of the Australian Mathematical Society , First View , pp. 1 - 22
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Dmitry Badziahin

References

Alniaçik, K., ‘On semi-strong U-numbers’, Acta Arith. 60(4) (1992), 349–358.CrossRefGoogle Scholar
Alniaçik, K. and Saias, E., ‘Une remarque sur les ${G}_{\delta }$ -denses’, Arch. Math. (Basel) 62(5) (1994), 425–426; a remark on ${G}_{\delta }$ -dense sets.CrossRefGoogle Scholar
Beresnevich, V. and Velani, S., ‘Schmidt’s theorem, Hausdorff measures, and slicing’, Int. Math. Res. Not. IMRN 2006 (2006), Article ID 48794, 24 pages.Google Scholar
Burger, E. B., ‘Diophantine inequalities and irrationality measures for certain transcendental numbers’, Indian J. Pure Appl. Math. 32 (2001), 1591–1599.Google Scholar
Drensky, V. and Formanek, E., Polynomial Identity Rings, Advanced Courses in Mathematics. CRM Barcelona (Birkhäuser Verlag, Basel, 2004).CrossRefGoogle Scholar
Erdős, P., ‘Representations of real numbers as sums and products of Liouville numbers’, Michigan Math. J. 9 (1962), 59–60.CrossRefGoogle Scholar
Hussain, M., Schleischitz, J. and Ward, B., ‘The folklore set and Dirichlet spectrum for matrices’, Preprint, 2024, arXiv:2402.13451.Google Scholar
Kleinbock, D. Y. and Margulis, G. A., ‘Flows on homogeneous spaces and Diophantine approximation on manifolds’, Ann. of Math. (2) 148(1) (1998), 339–360.CrossRefGoogle Scholar
Lelis, J. and Marques, D., ‘On transcendental entire functions mapping $\mathbb{Q}$ into itself’, J. Number Theory 206 (2020), 310–319.CrossRefGoogle Scholar
Lelis, J., Marques, D. and Moreira, C. G., ‘A note on transcendental analytic functions with rational coefficients mapping $\mathbb{Q}$ into itself’, Proc. Japan Acad. Ser. A Math. Sci. 100(8) (2024), 43–45.Google Scholar
Mahler, K., ‘Some suggestions for further research’, Bull. Aust. Math. Soc. 29(1) (1984), 101–108.CrossRefGoogle Scholar
Maillet, E., Introduction a la theorie des nombres transcendants et des proprietes arithmetiques des fonctions (Gauthier-Villars, Paris, 1906).Google Scholar
Marques, D. and Moreira, C. G., ‘On a variant of a question proposed by K. Mahler concerning Liouville numbers’, Bull. Aust. Math. Soc. 91(1) (2015), 29–33.CrossRefGoogle Scholar
Marques, D. and Ramirez, J., ‘On transcendental analytic functions mapping an uncountable class of U-numbers into Liouville numbers’, Proc. Japan Acad. Ser. A Math. Sci. 91(2) (2015), 25–28.CrossRefGoogle Scholar
Marques, D., Ramirez, J. and Silva, E., ‘A note on lacunary power series with rational coefficients’, Bull. Aust. Math. Soc. 93(3) (2016), 372–374.CrossRefGoogle Scholar
Marques, D. and Schleischitz, J., ‘On a problem posed by Mahler’, J. Aust. Math. Soc. 100(1) (2016), 86–107.CrossRefGoogle Scholar
Marques, D. and Silva, E., ‘A note on transcendental power series mapping the set of rational numbers into itself’, Commun. Math. 25(1) (2017), 1–4.CrossRefGoogle Scholar
Moshchevitin, N. G., ‘Singular Diophantine systems of A. Ya. Khinchin and their application’, Uspekhi Mat. Nauk 65(3(393)) (2010), 43–126 (in Russian); translation in Russian Math. Surveys 65(3) (2010), 433–511.Google Scholar
Nakagawa, J. and Horie, K., ‘Elliptic curves with no rational points’, Proc. Amer. Math. Soc. 104(1) (1988), 20–24.CrossRefGoogle Scholar
Oxtoby, J., Measure and Category. A Survey of the Analogies Between Topological and Measure Spaces, 2nd edn, Graduate Texts in Mathematics, 2 (Springer-Verlag, New York–Berlin, 1980).Google Scholar
Petruska, G., ‘On strong Liouville numbers’, Indag. Math. (N.S.) 3(2) (1992), 211–218.CrossRefGoogle Scholar
Rieger, G. J., ‘Über die Lösbarkeit von Gleichungssystemen durch Liouville–Zahlen’, Arch. Math. (Basel) 26 (1975), 40–43 (in German).CrossRefGoogle Scholar
Schleischitz, J., ‘Rational approximation to algebraic varieties and a new exponent of simultaneous approximation’, Monatsh. Math. 182(4) (2017), 941–956.CrossRefGoogle Scholar
Schwarz, W., ‘Liouville–Zahlen und der Satz von Baire’, Math. Phys. Semesterber. 24(1) (1977), 84–87 (in German).Google Scholar
Senthil Kumar, K., Thangadurai, R. and Waldschmidt, M., ‘Liouville numbers and Schanuel’s Conjecture’, Arch. Math. (Basel) 102(1) (2014), 59–70.CrossRefGoogle Scholar
Tricot, C. Jr, ‘Two definitions of fractional dimension’, Math. Proc. Cambridge Philos. Soc. 91(1) (1982), 57–74.CrossRefGoogle Scholar