Abstract
We establish a new transcendence criterion of p-adic continued fractions which are called Ruban continued fractions. By this result, we give explicit transcendental Ruban continued fractions with bounded p-adic absolute value of partial quotients. This is p-adic analogy of Baker’s result. We also prove that p-adic analogy of Lagrange theorem for Ruban continued fractions is not true.
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1 Introduction
Maillet [7] is the first person who gave explicit transcendental continued fractions with bounded partial quotients. After that, Baker [1] extended Maillet’s results with LeVeque Theorem [5] which is Roth Theorem for algebraic number fields.
There exist continued fraction expansions in p-adic number field \({\mathbb {Q}}_p\), not just in \({\mathbb {R}}\). Schneider [9] was motivated by Mahler’s work [6] and gave an algorithm of p-adic continued fraction expansion. In the same year, Ruban [8] also gave an different algorithm of p-adic continued fraction expansion. Ubolsri, Laohakosol, Deze, and Wang gave several transcendence criteria for Ruban continued fractions (see [3, 12, 14, 15]). The proofs are mainly based on the theory of p-adic Diophantine approximations. However, they only studied Ruban continued fractions with unbounded p-adic absolute value of partial quotients. In this paper, we study analogy of Baker’s transcendence criterion for Ruban continued fractions with bounded p-adic absolute value of partial quotients.
Let p be a prime. We denote by \(|\cdot |_p\) the valuation normalized to satisfy \(|p|_p =1/p\). A function \(\lfloor \cdot \rfloor _p\) is given by the following:
where \(\alpha = \sum _{n=m}^{\infty } c_n p^n ,\ c_n \in \{0,1, \ldots , p-1 \},\ m \in {\mathbb {Z}},\ c_m \not = 0 .\) The function is called a p-adic floor function. If \(\alpha \not = \lfloor \alpha \rfloor _p\), then we can write \(\alpha \) in the form
with \(\alpha _1 \in {\mathbb {Q}}_p\). Note that \( |\alpha _1|_p \ge p\) and \( \lfloor \alpha _1 \rfloor _p \not = 0\). Similarly, if \(\alpha _1 \not = \lfloor \alpha _1 \rfloor _p\), then we have
with \(\alpha _2 \in {\mathbb {Q}}_p\). We continue the above process provided \(\alpha _n \not = \lfloor \alpha _n \rfloor _p\). In this way, it follows that \(\alpha \) can be written in the form
For simplicity of notation, we write the continued fraction
\(\alpha _n\) is called the n-th complete quotient and either \(\lfloor \alpha \rfloor _p\) or \(\lfloor \alpha _n \rfloor _p\) is called a partial quotient. If the above process stopped in a certain step, then
is called a finite Ruban continued fraction. Otherwise, in the same way, we have
which is called an infinite Ruban continued fraction. As a remark, according to the fact that the Hensel expansion of a p-adic number is unique, we have the uniqueness of Ruban continued fraction expansions.
We define \(S_p = \{ \lfloor \alpha \rfloor _p \ | \ \alpha \in {\mathbb {Q}}_p \},\ S' _p =\{\lfloor \alpha \rfloor _p \ | \ |\alpha |_p \ge p \ \text{ for }\ \alpha \in {\mathbb {Q}}_p \}\). Let \((a_i)_{i\ge 0}\) be a sequence with \(a_0 \in S_p\) and \(a_i \in S' _p\) for all \(i\ge 1\), and \((n_i)_{i\ge 0}\) be an increasing sequence of positive integers. Let \((\lambda _i)_{i\ge 0}\) and \((k_i)_{i\ge 0}\) be sequences of positive integers. Assume that for all i,
Consider a p-adic number \(\alpha \) defined by
Then \(\alpha \) is called a quasi-periodic Ruban continued fraction.
The main theorem is the following.
Theorem 1
Let \((a_i)_{i\ge 0}, (n_i)_{i\ge 0}, (\lambda _i)_{i\ge 0}, \text{ and } (k_i)_{i\ge 0}\) be as in the above, and \(A\ge p\) be a real number. Assume that \((a_i)_{i\ge 0}\) is a non-ultimately periodic sequence such that \(|a_i |_p \le A\) for each i. If \(a_{n_i} = a_{n_i +1} = \cdots = a_{n_i + k_i -1} = (p-1)+(p-1)p^{-1}=p-p^{-1}\) for infinitely many i and
where B is defined by
then \(\alpha \) is transcendental.
As a remark, a sequence \((a_n)_{n \ge 0}\) is said to be ultimately periodic if there exist integers \(k\ge 0\) and \(l\ge 1\) such that \(a_{n+l} =a_{n}\) for all \(n\ge k\).
For example, the following p-adic numbers are transcendental:
where \(2\cdot 3^i\) and \(8\cdot 17^i\) indicate the number of times a block of partial quotients is repeated. (1) is the case that for \(i\ge 0,\ a_{n_{2i}}=p-p^{-1},\ a_{n_{2i-1}}=p^{-1},\ n_i=3^i ,\ \lambda _i=2\cdot 3^i ,\ k_i=1,\ A=p\) in Theorem 1. (2) is the case that for \(i\ge 0,\ a_{n_{2i}}=p^{-1},\ a_{n_{2i} +1}=p^{-2},\ a_{n_{2i+1}}=a_{n_{2i+1} +1}=p-p^{-1} ,\ n_i=17^i ,\ \lambda _i=8\cdot 17^i ,\ k_i=2,\ A=p^2\) in Theorem 1.
A well-known Lagrange’s theorem states that the continued fraction expansion for a real number is ultimately periodic if and only if the number is quadratic irrational. For Schneider continued fractions, p-adic analogy of Lagrange’s theorem is not true, that is, there exists a quadratic irrational number whose Schneider continued fraction is not ultimately periodic (See e.g. Weger [2], Tilborghs [11], van der Poorten [13]). This paper deals with analogy of Lagrange’s theorem for Ruban continued fractions.
We prove that analogy of Lagrange’s theorem for Ruban continued fractions is not true in Sect. 2. Auxiliary results for main results are gathered in Sect. 3. In Sect. 4, we prove Theorem 1 and give criteria of quadratic or transcendental in a certain class of Ruban continued fractions. These proofs are mainly based on the proof of Baker’s results and the non-Archimedean version of Roth’s theorem for an algebraic number field [10].
2 Rational and quadratic irrational numbers
Wang [14] and Laohakosol [4] characterized rational numbers with Ruban continued fractions as follows.
Proposition 2
Let \(\alpha \) be a p-adic number. Then \(\alpha \) is rational if and only if its Ruban continued fraction expansion is finite or ultimately periodic with the period \(p-p^{-1}\).
Proof
Next, we prove that analogy of Lagrange’s theorem for Ruban continued fractions is not true by the similar method as in [2]. We consider a Ruban continued fraction for \(\alpha = \sqrt{D}\) where \(D \in {\mathbb {Z}}\) not a square, but a quadratic residue modulo p, if p is odd, 1 modulo 8, if \(p=2\), so that \(\alpha \in {\mathbb {Q}}_p\). If the Ruban continued fraction of \(\alpha \) is \([a_0 , a_1 , a_2 , \ldots ]\), then there exist rational numbers \(R_n , Q_n\) such that
for \(n \in {\mathbb {Z}}_{\ge 0}\). Obviously, \(R_0 = 0, Q_0 = 1\), and for all n we have the recursion formula
by induction on n.
Proposition 3
If \(\ R_m Q_m \le 0,\ \text{ and }\ R_{m+1}^2 >D\ \text{ for } \text{ some }\ m\), then the Ruban continued fraction expansion of \(\alpha \) is not ultimately periodic.
Proof
We show \(R_{m+1} Q_{m+1} < 0,\ R_{m+2}^2>D,\ \text{ and }\ |R_{m+2}|>|R_{m+1}|\). Let us assume \(R_m Q_m < 0\). Then we have \(R_m R_{m+1} < 0\) by the recursion formula for \(R_{m+1}\). We also obtain \(Q_m Q_{m+1} < 0\) by the recursion formula for \(Q_{m+1}\) and \(R_{m+1} ^2 > D\). Hence, we get \(R_{m+1} Q_{m+1} <0\). Furthermore, by \(a_{m+1} \not = 0\), we have
so that \(R_{m+2} ^2 > D\). Next, let us assume \(R_m Q_m = 0\). By \(R_{m} =0\), we have \(R_{m+1} = a_m Q_m\). By the recursion formula for \(Q_{m+1}\), we have \(Q_{m} Q_{m+1}<0\). Thus, we obtain \(R_{m+1} Q_{m+1} < 0\). In the same way, we see \(|R_{m+2}| > |R_{m+1}|\) and \(R_{m+2}^2 >D\). Since \((|R_n|)_{n \ge m}\) is strictly increasing, the Ruban continued fraction expansion for \(\sqrt{D}\) is not ultimately periodic. \(\square \)
Corollary 4
If \(D<0\), then the Ruban continued fraction expansion of p-adic number \(\sqrt{D}\) is not ultimately periodic.
Proof
Since \(R_0 Q_0 =0\), and \(R_1 ^2 \ge 0\), the corollary follows. \(\square \)
3 Auxiliary results
For an infinite Ruban continued fraction \(\alpha =[a_0 , a_1 , a_2 , \ldots ]\), we define nonnegative rational numbers \(q_n , r_n\) by using recurrence equations:
Let \(\lambda \) be a variable. Then the Ruban continued fraction has the following properties which are the same properties as the continued fraction expansions for real numbers: For all \(n\ge 0\),
Those are easily seen by induction on n.
Lemma 5
The following equalities hold:
Proof
See [14]. \(\square \)
Lemma 6
If \(\alpha '\) is a Ruban continued fraction in which the first \(n+1\) partial quotients are the same as those of \(\alpha \), then
Proof
Since \(r_n/q_n\) is a n-th convergent to both \(\alpha \) and \(\alpha '\), and (8), the lemma follows. \(\square \)
Lemma 7
The following inequalities hold:
Proof
The proof is by induction on n. It is obvious that for \(n=-1,0\). By Lemma 5 and the definition of Ruban continued fraction expansions, we have
The proof for \(r_n\) is similar. \(\square \)
For \(\beta \in \overline{{\mathbb {Q}}}\), let \(f_\beta (X)= \sum _{i=0}^{n} d_i X^i\) be a minimum polynomial of \(\beta \) in \({\mathbb {Z}}[X]\). Put
\(H(\beta )\) is called a primitive height of \(\beta \).
Lemma 8
Suppose \(a_0 =0\). Let h, k be positive integers and consider the Ruban continued fraction
Then \(\eta \) is rational or quadratic irrational. Furthermore, we have
Proof
By \(\eta _h = \eta _{h+k}\) and (4), we obtain
Eliminating \(\eta _h\), we have
where
Therefore, \(\eta \) is either rational or quadratic irrational. By the assumption \(a_0 =0\), it follows that \(r_n \le q_n ,\ |r_n|_p \le |q_n|_p \ \text{ for } \text{ all }\ n\ge 0\). By induction on n, it is easy to check that \(r_n|r_n|_p ,\ q_n|q_n|_p \in {\mathbb {Z}}\ \text{ for } \text{ all }\ n\ge 0\).
Let us assume that \(\eta \) is a quadratic irrational. By \(|q_{h+k-1}|^2 _p A\), \(|q_{h+k-1}|^2 _p B\), \(|q_{h+k-1}|^2 _p C \in {\mathbb {Z}}\) and Lemma 7, we obtain
Now let us assume that \(\eta \) is rational. By Proposition 2, we have
that is,
When \(h=1\), we see that \(H(\eta )=p\). Next we consider the case \(h\ge 2\). Since \((p r_{h-2} - r_{h-1})|q_{h-1}|_p\) and \((p q_{h-2} - q_{h-1})|q_{h-1}|_p\) are integers, we have
and the lemma follows. \(\square \)
We recall a height of algebraic numbers which is different from the primitive height. Let K be an algebraic number field and \({\mathcal {O}}_K\) be the integer ring of K, and M(K) be the set of places of K. For \(x \in K\) and \(v \in M(K)\), we define the absolute value \(|x|_v\) by
-
(i)
\(|x|_v = |\sigma (x)|\) if v corresponds the embedding \(\sigma : K \hookrightarrow {\mathbb {R}}\)
-
(ii)
\(|x|_v = |\sigma (x)|^2 = |{\overline{\sigma }} (x)|^2\) if v corresponds the pair of conjugate embeddings \(\sigma , {\overline{\sigma }} : K \hookrightarrow {\mathbb {C}}\)
-
(iii)
\(|x|_v = ( N ({\mathfrak {p}}))^{- ord _{{\mathfrak {p}}} (x)}\) if v corresponds to the prime ideal \({\mathfrak {p}}\) of \({\mathcal {O}}_K\).
Set
for \(\beta \in K\). \({\overline{H}}_K (\beta )\) is called an absolute height of \(\beta \). Then there are the following relations between primitive and absolute height.
Proposition 9
For \(b \in {\mathbb {Q}}\) and \(\beta \in \overline{{\mathbb {Q}}}\) with \([{\mathbb {Q}}(\beta ) , {\mathbb {Q}}] = D\), we have
Proof
See Part B of [10]. \(\square \)
The main tool for the proof of main results is the non-Archimedean version of Roth’s theorem for algebraic number fields.
Theorem 10
(Roth Theorem) Let K be an algebraic number field, and v be in M(K) with it extended in some way to \({\overline{K}}\). Let \(\beta \in {\overline{K}} \backslash K\) and \(\delta ,C >0\) be given. Then there are only finite many \(\gamma \in K\) with the solution of the following inequality:
Proof
See Part D of [10]. \(\square \)
4 Main results
Proof of Theorem 1
We may assume that \(a_0=0\). By the assumption, there are infinitely many positive integers j which satisfy
Let \(\varLambda \) be an infinite set of j which satisfy (9).
For \(i\in \varLambda \), we put
By Proposition 2, \(\alpha \) is not rational. Suppose that \(\alpha \) is an algebraic number of degree at least two. We show that if \(\chi >2\), then we have
for all sufficiently large \(i \in \varLambda \). Suppose the claim is false. By Proposition 2, \(\eta ^{(i)}\) is rational for each \(i \in \varLambda \). By Lemma 8 and Proposition 9, we have
for infinitely many i, which contradicts Theorem 10.
By Lemma 6, we obtain \(|\alpha - \eta ^{(i)}|_p \le |q_{m_i}|_p ^{-2}\) for \(i \in \varLambda \), where \(m_i =n_i +k_i \lambda _i -1\). Therefore, we get
for sufficiently large \(i \in \varLambda \). By Lemma 5, we see \(p^i \le |q_i|_p \le A^i\) for \(i\ge 1\). Thus, for all sufficiently large \(i \in \varLambda \), it follows that
Since there exists \(\delta >0\) such that \(\lambda _i >(B +\delta )n_i\) for all sufficiently large i, we have for all sufficiently large \(i \in \varLambda \),
This inequality holds for each \(\chi >2\), a contradiction. \(\square \)
We also obtain the following results.
Theorem 11
Let \(\alpha \) be a quasi-periodic Ruban continued fraction, and \(A\ge p\) be a real number. Assume that \((a_i)_{i\ge 0}\) is a non-ultimately periodic sequence such that \(|a_i |_p \le A\) for each i, and \((k_i)_{i\ge 0}\) is bounded. If
where \(B'\) is defined by
then \(\alpha \) is quadratic irrational or transcendental.
Theorem 12
Consider a quasi-periodic Ruban continued fraction
where the notation means that \(n_i = n_{i-1} + \lambda _{i-1}k_{i-1}\). Assume that \((a_i)_{i\ge 0}\) is not an ultimately periodic sequence, the sequences \((|a_i|_p)_{i\ge 0} \ \text{ and }\ (k_i)_{i\ge 0}\) are bounded, and that
Then \(\alpha \) is quadratic irrational or transcendental.
Remark 13
There exist quadratic irrational numbers whose Ruban continued fraction expansions are not ultimately periodic by Corollary 4. Therefore, it is difficult to determine whether a given Ruban continued fraction is quadratic irrational or transcendental. However, we see that there exist a transcendental number in the set of Ruban continued fractions which satisfy the assumption of Theorem 11 and 12. For example, (2) satisfies the assumption of Theorem 11 and 12.
In the following, \(c_1 , c_2 , \ldots ,c_6\) denote positive real numbers which depend only on \(\alpha \), and we may assume that \(a_0 =0\).
Proof of Theorem 11
By the assumption, there exists \(\delta >0\) such that \(\lambda _i >(B'+\delta )n_i\) for infinitely many i. For each positive integer i, there are only finitely many possibilities for \(k_i\) and for
Therefore, there exist a positive integer k and \(b_1 ,b_2 , \ldots ,b_k \in S' _p\) such that there are infinitely many j which satisfy
Let \(\varLambda \) be an infinite set of j which satisfy (11).
For \(i\in \varLambda \), we put
By Proposition 2, \(\alpha \) is not rational. Suppose that \(\alpha \) is an algebraic number of degree at least three. We show that if \(\chi >2\), then we have
for all sufficiently large \(i \in \varLambda \). Suppose the claim is false. By Lemma 8, \(\eta ^{(i)}\) is rational or quadratic irrational for each \(i \in \varLambda \). Let us assume that \(\eta ^{(i)}\) is quadratic irrational. Then there exists a quadratic field K such that \(\eta ^{(i)} \in K\) for all \(i \in \varLambda \). Take a real number \(\varepsilon \) which satisfies \(0< \varepsilon < \chi -2\). Then we have \(2^{\chi -\varepsilon } < |q_{n_i + k_i -1}|_p ^{4\varepsilon }\) for all sufficiently large \(i \in \varLambda \). Put \(v \in M(K)\) with \(v \mid p\). We denote again by v one of the place extended to \(K(\alpha )\). By \([K(\alpha )_v : {\mathbb {Q}}_p]=1\), Lemma 8, and Proposition 9, we obtain
for infinitely many i, which contradicts Theorem 10. In the same way, we see (12) in the case that \(\eta ^{(i)}\) is rational. By Lemma 6, we have \(|\alpha - \eta ^{(i)}|_p \le |q_{m_i}|_p ^{-2}\) for \(i \in \varLambda \), where \(m_i =n_i +k\lambda _i -1\). Therefore, we obtain
for sufficiently large \(i \in \varLambda \). By Lemma 5, we see \(p^i \le |q_i|_p \le A^i\) for \(i\ge 1\). Thus, for all sufficiently large \(i \in \varLambda \), we have
so
This inequality holds for each \(\chi >2\), and contradicts if i is sufficiently large in \(\varLambda \). \(\square \)
Proof of Theorem 12
Put
for \(i=0,1,2,\ldots \ \text{ and }\ h=1,2,\ldots ,k_i\). Put
For each positive integer i, there exist only finitely many possibilities for \(k_i\) and for
\(P^{(i)}\) is a function which depends only on \(k_i , a_{n_i}, a_{n_i +1}, \ldots , a_{n_i +k_i -1}\). Hence, there exists a real number P such that the greatest of those values \(|P^{(i)}|_p\) which are attained for infinitely many i. Then there exists an integer l such that
There exist a positive integer k and \(b_1 ,b_2 , \ldots ,b_k \in S' _p\) such that there are infinitely many j which satisfy
Let \(\varLambda \) be an infinite set of j which satisfy (13). We may assume that \(l=0\).
Let us show that
Firstly, an induction allows us to establish the mirror formula
Put
for \(i=0,1,2, \ldots \ \text{ and }\ h=1,2, \ldots ,k_i \lambda _i\), and
Clearly, we have \(q_{n_{i+1} -1} =W^{(i)} q_{n_i -1}\). It follows from Lemma 5 and 6 that for any i,
where \(U_{1,0} ^{(i)} =1\) and otherwise \(U_{h,s} ^{(i)}\) is the denominator of \((h+s k_i -1)\)-th convergent to \(P_{h} ^{(i)}\). Likewise, for all i, we have
If \(i \in \varLambda \), then \(|P^{(i)}|_p = P\) and \(P_1 ^{(i)}\) is independent of i. Therefore, we obtain
If A and K are the upper bounds of the sequences \((|a_i|_p)_{i\ge 0}\) and \((k_i)_{i\ge 0}\), then for all i, we have
Now, there exist a real number \(\delta >0\) and an integer \(N\ge 1\) such that \(\lambda _i >(4+\delta )\lambda _{i-1}\) for all \(i>N\). Set \(\chi :=2 + \delta /4\). For \(i\in \varLambda \), we put
By Proposition 2, \(\alpha \) is not rational. Suppose that \(\alpha \) is an algebraic number of degree at least three. Then we have
for all sufficiently large \(i \in \varLambda \). This follows by the same way as in the proof of Theorem 11. By Lemma 6, we see \(|\alpha - \eta ^{(i)}|_p \le |q_{n_{i+1} -1}|_p ^{-2}\) for all i. Therefore, we obtain
for all sufficiently large \(i \in \varLambda \). Applying (14), (15), (16), and (17), we have for all sufficiently large \(i \in \varLambda \),
Taking logarithms, we see that for all sufficiently large \(i \in \varLambda \),
Since \(i/\lambda _i \rightarrow 0\ \text{ as }\ i \rightarrow \infty \), we have
for all sufficiently large \(i \in \varLambda \). This contradicts, and the proof is complete. \(\square \)
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Acknowledgements
I am greatly indebted to Prof. Kenichiro Kimura for several helpful comments concerning the proof of main theorems. I also wish to express my gratitude to Prof. Masahiko Miyamoto.
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Ooto, T. Transcendental p-adic continued fractions. Math. Z. 287, 1053–1064 (2017). https://doi.org/10.1007/s00209-017-1859-2
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DOI: https://doi.org/10.1007/s00209-017-1859-2