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IRRATIONALITY EXPONENTS OF NUMBERS
RELATED WITH CAHEN’S CONSTANT

IEKATA SHIOKAWA

This is a report on recent results of myself jointly with Duverney [7] on irra‐tionality exponents of numbers related with Cahen^{:}s constant.

For a real number \alpha , the irrationality exponent \mu(\alpha) is defined by the greatest
lower bound of the set of numbers \mu for which the inequality(0.1)| \alpha-\frac{p}{q}|<\frac{1}{q^{\mu}has only finitely many rational solutions p/q , or equivalently the least upper boundof the set of numbers \mu for which the inequality (0.1) has infinitely many solutions.
If \alpha is irrational, then \mu(\alpha)\geq 2 . If \alpha is a real algebraic irrationality, then \mu(\alpha)=2by Roth’s theorem. If l^{4}((y)=\infty , then cr is called a Liouville number.
The main theorem of this paper, [7] stated below gives lower and upper bounds
for the irrationality exponents of continued fractions.
We employ the usual notations for continued fractions :b_{0}+ \frac{a_{1} {b_{1} \frac{a_{2} {b_{2} + \cdot\cdot\cdot +^{\frac{a_{n} {b_{n} =b_{0}+\frac{a_{1} {b_{1}+\frac+{a\_fr{a2c{}a_{{bn}_{{b2_}{n+}} =\frac{A_{n} {B_{n} ,.

and\underline{A_{n} ,\underline{a_{2}\underline{a_{3}\limb_{0}+ —aı
b_{1}+b_{2}+b_{3}+ narrow+\infty B_{n}=where \{A_{\uparrow l}\} and \{B_{n}\} are defined by(0.2)\begin{ar y}{l A_{-1}=1,A_{0}=b_{0},B_{-1}=0,B_{0}=1, A_{n}=b_{n}A_{n-1}+a_{n}A_{n-2}(n\geq1) B_{n}=b_{n}B_{n-\imath}+a_{n}B_{n-2}(n\geq1) \end{ar y}

Theorem 1. Let an infinite continued fraction\alpha=\underline{a_{1} \underline{a_{2} \underline{a_{3}(0.3)

b_{1}+b_{2}+b_{3}+...

be convergent, where a_{n} , b_{n}(n\geq 1) are non zero rational integers. Assume that

(0.4)

\sum_{n={\imath} ^{+\infty}|\frac{a_{n+1} {b_{n}b_{n+1} |<\infty,(0.5)\lim_{nar ow+\infty}|\frac{a_{1}a_{2}\cdots a_{n} {b_{1}b_{2}\cdots b_{n} |=0.

and2010 Mathe7nati_{Cb}b^{\tau}ub)ectC,la6 sification. Primary
Key words and phrases. Irrationality exponent,
Sylvester sequence, Sierpinski sequence.

11J82 ; SecondaryllJ70.
Cahen’s constant, Continued fractions,

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IEKATA SHIOKAWA

Then \alpha is irrational and its irrationality exponent \mu(\alpha) satisfies(0.6)

2+ \sigma\leq\mu(\alpha)\leq 2+\max(\tau_{1}, \tau_{2}) ,where\sigma=1\dot{ \imath} m\sup_{nar ow+\infty}\frac{\log|b_{n+1}|-\log|a_{1}a_{2} \cdots a_{n+1}| {\log|b_{1}b_{2}\cdots b_{n}| ,ı=1 \dot{ \imath} m\sup_{nar ow+\infty}\frac{1og.|a_{1}a_{2}\cdots a_{n+{\imath} }| {iog|b_{1}b_{2}\cdot\cdot b_{n}|-\log|a_{ \imath} a_{2}\cdots a_{n}| ,

(0.7)
(0.8)

\mathcal{T}and\tau_{2}=\lim_{nar ow+}\sup_{\infty}\frac{\log|b_{n+l}|-\log|a_{1}a_{2}\cdots a_{n+{\imath} |+2.1.og(A_{n},B_{n}) {\log|b_{1}b_{2}\cdots b_{7?}|- \log|a_{ \imath} a_{2}\cdot a_{?l}|with (A_{n}, B_{n}) the greatest common divisor of A_{n} and B_{n}.

We apply Theorem 1 to continued fractions representing numbers related toCahen’s constant and deduce their transcendence from the obtained lower boundsof their irrationality exponents.

In 1880 Sylvester [11] proved that any real number

0<x<1 can be expanded

uniquely in the series

x=\sum_{n=0}^{+\infty}\frac{1}{t_{n} ,where the t_{n} are integers satisfying the condition t_{0}\geq 2, t_{n+{\imath}}\geq t_{n}^{2}-t_{n}+1(n\geq 0) ,and furthermore that x is irrational if and only if the equality holds for all large n.

He examined some of the properties of the (Sylvester) sequence \{S_{n}\}_{?\not\supset\geq 0} definedby(0.9)

S_{0}=2,S_{n+{\imath}}=S_{n}^{2}-S_{n}+ ı(n\geq 0) ,which satisfies\sum_{n=0}^{+\infty}\frac{1}{S_{n}=ı.

Cahen [2] and Sierpinski [9] independently obtained similar results for alternatingseries; namely, any irrational number

0<x<1 can be uniquely written in the form

x= \sum_{n=0}^{+\infty}\frac{(-1)^{n} {u_{n} ,where the u_{n} are integers satisfying u_{0}\geq ı, u_{n+1}\geq u_{7?}^{2}+u_{n}(n\geq 0) . As an example,

Cahen [2] mentioned that (Cahen’s constant)\sum_{n=0}^{+\infty}\frac{(-1)^{n} {u_{n} =\frac{1}{1}-\frac{1}{2}+\frac{1}{6} -\frac{1}{42}+\frac{1}{1806}-\frac{1}{32634 2}+

is an irrational number, where u_{0}=1, u_{r\iota+1}=u_{n}^{2}+u_{n}(n\geq 0) , and hence
S_{n}- ı (n\geq 0) . We note that the sequence \{s_{n}\}_{n\geq 0} defined by(0.10)

s_{0}=2, s_{n+1}=s_{n}^{2}+S_{?1}-1 (n\geq 0)

satisfies\sum_{n=0}^{+\infty}\frac{(-1)^{n} {s_{n} =\frac{1}{3}.

u_{n}=

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NUMBERS RELATED WITH CAHEN’S CONSTANTIn 199ı Davison and Shallit [4] proved the transcendence of Cahen’s constant.
Becker [1] generalized the result by Mahler’s method.
In this paper we generalize the sequences S_{n} and s_{n} defined in (0.9) and (0.10)
by introducing the sequences u_{n}=u_{n}(\varepsilon) satisfying u_{0}\in \mathbb{N}, u_{0}> \max(1, \varepsilon) , andthe recurrence

u_{n+1}=u_{n}^{2}-eu_{n}+\varepsilon (n\geq 0) ,(0.11)

where \varepsilon is a non‐zero integer given arbitrary. Next, we define the numbers \gamma_{l,\varepsilon}=\gamma_{l,\in}(u_{0}) by(0.12)\gam a_{l,\varepsilon}=\sum_{n=0}^{+\infty} (- {\imath})^{n} (\frac{\varepsilon^{n} {u_{\eta}-\varepsilon})^{l} (l=1,23_{:}\cdots).

We expand the numbers \gamma_{l,\varepsilon} in continued fractions whose partial numerators a_{n}and denominators b_{n} satisfy the assumptions in Theorem 1. Applying Theorem 1,
we obtain the following

Theorem 2. Let \gamma_{l,\varepsilon} be the n\uparrow lmbcr.sd\supset

ed b.y (0.12).

A.s.s?ime that u_{0}and \varepsilon are

coprime. Then \mu(\gamma_{{\imath},\varepsilon})=3 and(0.13)

2+ \frac{2}{3l-1}\leq\mu(\gamma_{l,\in})\leq 2+\frac{6(l-1)}{3l+1} (l=2,3,4, \cdots) .

Corollary 1. For every positive integer l_{:}\gamma_{l,\in} is a non‐Liouville transcendental
number.

We give some examples of the numbers \gamma_{l,\varepsilon}.
Example 1. When \varepsilon=1 and u_{0}=2 , we have\gamma_{l,1}(2)=\sum_{n=0}^{+\infty}\frac{(-1)^{n} {(S_{n}-1)^{l} (l=1,2,3, \cdots).

In particular, \gamma_{1,1}(2) is Cahen’s constant.
Example 2. When \varepsilon=-1 and u_{0}=2 , we obtain\gamma_{l,-1}(2)=\sum_{n=0}^{+\infty}\frac{(-1)^{n} {(s_{n}+{\imath})^{l} (l= 2,4,6, \cdots)= \sum_{n=0}^{+\infty}\frac{1}{(s_{n}+1)^{l} (l=1,3,5, \cdots)\gamma_{l} ,‐ı (2),.

Example 3. When \varepsilon=2 and u_{0}=3, u_{n} is the n‐th Fermat number:u_{7?}=F_{n}=2^{2^{n}}+1.
Therefore we have\gamma_{l,2}(3)=\sum_{n=0}^{+\infty}(-1)^{n}(\frac{2^{n} {F_{n}-2})^{l}= \sum_{n=0}^{+\infty}(-1)^{n}(\frac{2^{n} {2^{2^{n} -1})^{l} (l=1,2,3, \cdots).

It should be noted that the irrationality exponent of the sum of the reciprocalsof Fermat numbers is equal to 2 (see [3]).

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IEKATA SHIOKAWA

Example 4. Denote by L_{n} the sequence of Lucas numbers. Definev_{n}=L_{2^{n+1}}=\Phi^{2^{n+1}}+\Phi^{-2^{n+1}}
where \Phi=\frac{1}{2}(1+\sqrt{5}) is the Golden number. Then clearly
put u_{n}=v_{n}+2 , we see that u_{0}=5 andv_{n+1}=v_{n}^{2}-2 . If we

u_{n+{\imath}}=u_{n}^{2}-4u_{n}+4
for every n\geq 0 . Therefore\gamma_{l,4}(5)=\sum_{n=0}^{+\infty}(-1)^{n}(\frac{4^{n} {L_{2^{71+1}}-2})^{l} (l=1,2_{:}3_{:}\cdots).

As for transcendence, much more general results are obtained by Duverney, Kuro‐sawa, and Shiokawa [6]. They discussed transcendence of the values of the series\sum_{n=0}^{\infty}\frac{a^{n} {q(p^{n}(z) }at algebraic points, where a\in\overline{\mathbb{Q} ,p(z) , q(z)\in\overline{\mathbb{Q} [z] with \deg p(z)\geq 2 and \deg q(z)\geq
1 . For example, [6, Example 1.5] states that, if a\neq 0 and \gamma with S_{n}\neq\gamma for alln\geq 0 are algebraic numbers, then\sum_{n=0}^{\infty}\frac{a^{n}{(S_{n}-\gam a)^{l},where l is any positive integers, is algebraic if and only if

a=l=1 and\gamma=0.

REFERENCES[1] P.‐G. Becker, Algebraic independence of the values of certain series by Mahler’s method, Mh.
Math. 114 (1992), 183‐198.
[2] E. Cahen, Note sur un développement des quantités numériques, qui présente quelque analogie
avec celui en fraction continue, Nouv. Ann. Math. 10 (1891), 508‐514.
[3] M. Coons, On the rational approximation of the sum of the reciprocals of the Fermat numbers,
Ramanujan J. 30, Nol (2013), 39‐65; addendum ibid. 37, Nol (2015), 109‐111.
[4] J. L. Davison and J. O. Shallit, Continued fractions for some alternating series, Mh. Math.
111(1991) , ı19‐126.
[5] D. Duverney, Number Theory : an elementary introduction through diophantine problems,
Monographs in Number Theory 4, World Scientific, 2010.

[6] D. Duverney, T. Kurosawa, and I. Shiokawa, Transcendence of numbers related with Cahen’sconstant, to appear in Moscow J. of Cobinatrics and Number Theory.

[7] D. Duverney and I. Shiokawa, Irrationality exponents of numbers related with Cahen’s con‐stant, submitted , Monographs in Number Theory 4, World Scientific, 2010.

[8] W. B. Jones and W. J. Thron, Continued Fractions : analytic theory and applications, Ency‐
clopedia of Math. and its applications 11, Addison‐Wesley, 1980.

[9] W. Sierpinski, Sur un algorithme pour développer les nombres réels en séries rapidement
convergentes, Bull. Intern. Acad. Sci. Cracovie (1911), 508‐514.
[10] J. Sondow, Irrationality measures, irrationality bases and a theorem of Jarnik,
https://arxiv.org/abs/math/0406300v1 (2004).
[11] J. J. Sylvester, On a point in the theory of vulgar functions, Amer. J. Math. 3 (1880), 332‐335.
IEKATA SIIIOKAWA,

E‐mail address: shiokawa@beige.ocn.ne.jp