On the $b$-ary expansion of a real number whose irrationality exponent is close to $2$
DOI:
https://doi.org/10.2140/mscand.2026.132.319Keywords:
rational approximation, exponent of approximation, combinatorics on wordsAbstract
Let $b \ge 2$ be an integer and $\xi$ an irrational real number. We establish that, if the irrationality exponent of $\xi$ is less than $2.324 \dots$, then the $b$-ary expansion of $\xi$ cannot be “too simple”, in a suitable sense. This improves the results of our previous paper (Ann. Sc. Norm. Super. Pisa Cl. Sci. 17:3, (2017), 931–947).
References
B. Adamczewski, “On the expansion of some exponential periods in an integer base”, Math. Ann. 346:1 (2010), 107–116. https://doi.org/10.1007/s00208-009-0391-z
B. Adamczewski and Y. Bugeaud, “On the complexity of algebraic numbers, I: expansions in integer bases”, Ann. of Math. (2) 165:2 (2007), 547–565. https://doi.org/10.4007/annals.2007.165.547
B. Adamczewski and Y. Bugeaud, “Nombres réels de complexité sous-linéaire: mesures d'irrationalité et de transcendance”, J. Reine Angew. Math. 658 (2011), 65–98. https://doi.org/10.1515/CRELLE.2011.061
J.-P. Allouche and J. Shallit, Automatic sequences: theory, applications, generalizations, Cambridge Univ. Press, 2003. https://doi.org/10.1017/CBO9780511546563
V. Berthé, C. Holton, and L. Q. Zamboni, “Initial powers of Sturmian sequences”, Acta Arith. 122:4 (2006), 315–347. https://doi.org/10.4064/aa122-4-1
Y. Bugeaud and J.-H. Evertse, “On two notions of complexity of algebraic numbers”, Acta Arith. 133:3 (2008), 221–250. https://doi.org/10.4064/aa133-3-3
Y. Bugeaud and D. H. Kim, “On the b-ary expansions of log$(1+1/a)$ and $e$”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17:3 (2017), 931–947. https://journals.sns.it/index.php/annaliscienze/article/view/519
Y. Bugeaud and D. H. Kim, “A new complexity function, repetitions in Sturmian words, and irrationality exponents of Sturmian numbers”, Trans. Amer. Math. Soc. 371:5 (2019), 3281–3308. https://doi.org/10.1090/tran/7378
Y. Bugeaud and L. Liao, “Uniform Diophantine approximation related to $b$-ary and $β$-expansions”, Ergodic Theory Dynam. Systems 36:1 (2016), 1–22. https://doi.org/10.1017/etds.2014.66
S. Ferenczi and C. Mauduit, “Transcendence of numbers with a low complexity expansion”, J. Number Theory 67:2 (1997), 146–161. https://doi.org/10.1006/jnth.1997.2175
N. J. Fine and H. S. Wilf, “Uniqueness theorems for periodic functions”, Proc. Amer. Math. Soc. 16 (1965), 109–114. https://doi.org/10.2307/2034009
N. P. Fogg, Substitutions in dynamics, arithmetics and combinatorics, edited by V. Berthé et al., Lecture Notes in Math. 1794, Springer, 2002. https://doi.org/10.1007/b13861
M. Lothaire, Algebraic combinatorics on words, Encycl. Math. Appl. 90, Cambridge Univ. Press, 2002. https://doi.org/10.1017/CBO9781107326019
D. Zeilberger and W. Zudilin, “The irrationality measure of π is at most 7.103205334137…”, Mosc. J. Comb. Number Theory 9:4 (2020), 407–419. https://doi.org/10.2140/moscow.2020.9.407
