Mahler's classification and a certain class of p -adic numbers
aa r X i v : . [ m a t h . N T ] J un MAHLER’S CLASSIFICATION AND A CERTAIN CLASS OF p -ADICNUMBERS TOMOHIRO OOTO
Abstract.
In this paper, we study a relation between digits of p -adic numbers andMahler’s classification. We show that an irrational p -adic number whose digits are au-tomatic, primitive morphic, or Sturmian is an S -, T -, or U -number in the sense ofMahler’s classification. Furthermore, we give an algebraic independence criterion for p -adic numbers whose digits are Sturmian. Introduction
Let p be a prime. We denote by | · | p the absolute value of the p -adic number field Q p normalized to satisfy | p | p = 1 /p . We denote by ⌊ x ⌋ the integer part and ⌈ x ⌉ the upperinteger part of a real number x . We set P := { , , . . . , p − } .Let A be a finite set. Let A ∗ , A + , and A N denote the set of finite words over A , the setof nonempty finite words over A , and the set of infinite words over A , respectively. Wedenote by | W | the length of a finite word W over A . For any integer n ≥
0, write W n = W W . . . W ( n times repeated concatenation of the word W ) and W = W W . . . W . . . (infinitely many times repeated concatenation of the word W ). Note that W is equal tothe empty word. More generally, for any real number w ≥
0, write W w = W ⌊ w ⌋ W ′ , where W ′ is the prefix of W of length ⌈ ( w − ⌊ w ⌋ ) | W |⌉ . Let a = ( a n ) n ≥ be a sequence over theset A . We identify a with the infinite word a a . . . a n . . . . An infinite word a over A issaid to be ultimately periodic if there exist finite words U ∈ A ∗ and V ∈ A + such that a = U V .We recall the definition of automatic sequences, primitive morphic sequences, and Stur-mian sequences. Let k ≥ k -automaton is a sextuplet A = ( Q, Σ k , δ, q , ∆ , τ ) , where Q is a finite set, Σ k = { , , . . . , k − } , δ : Q × Σ k → Q is a map, q ∈ Q , ∆ is afinite set, and τ : Q → ∆ is a map. For an integer n ≥
0, we set W n := w m w m − . . . w ,where P mi =0 w i k i is the k -ary expansion of n . For q ∈ Q , we define recursively δ ( q, W n ) by δ ( q, W n ) = δ ( δ ( q, w m w m − . . . w ) , w ). A sequence a = ( a n ) n ≥ is said to be k -automatic if there exists a k -automaton A = ( Q, Σ k , δ, q , ∆ , τ ) such that a n = τ ( δ ( q , W n )) for all n ≥
0. A sequence a = ( a n ) n ≥ is said to be automatic if there exists an integer k ≥ a is k -automatic.Let A and B be finite sets. A map σ : A ∗ → B ∗ is said to be a morphism if σ ( U V ) = σ ( U ) σ ( V ) for all U, V ∈ A ∗ . We define the width of σ by max a ∈A | σ ( a ) | . We say that σ is k -uniform if there exists an integer k ≥ | σ ( a ) | = k for all a ∈ A . In particular,we call a 1-uniform morphism a coding . A morphism σ : A ∗ → A ∗ is said to be primitive if there exists an integer n ≥ a occurs in σ n ( b ) for all a, b ∈ A . A morphism σ : A ∗ → A ∗ is said to be prolongable on a ∈ A if σ ( a ) = aW where W ∈ A + , and σ n ( W ) Mathematics Subject Classification.
Key words and phrases.
Mahler’s classification; automatic sequences. is not empty word for all n ≥
1. We say that a sequence a = ( a n ) n ≥ is primitive morphic if there exist finite sets A , B , a primitive morphism σ : A ∗ → A ∗ which is prolongable onsome a ∈ A , and a cording τ : A ∗ → B ∗ such that a = lim n →∞ τ ( σ n ( a )).Let 0 < θ < ρ be a real number. For an integer n ≥
1, we put s n,θ,ρ := ⌊ ( n + 1) θ + ρ ⌋ − ⌊ nθ + ρ ⌋ and s ′ n,θ,ρ := ⌈ ( n + 1) θ + ρ ⌉ − ⌈ nθ + ρ ⌉ .Note that s n,θ,ρ , s ′ n,θ,ρ ∈ { , } . We also put s θ,ρ := ( s n,θ,ρ ) n ≥ and s ′ θ,ρ := ( s ′ n,θ,ρ ) n ≥ . Asequence a = ( a n ) n ≥ is called Sturmian if there exist an irrational real number 0 < θ < ρ , a finite set A , and a coding τ : { , } ∗ → A ∗ with τ (0) = τ (1) suchthat ( a n ) n ≥ = ( τ ( s n,θ,ρ )) n ≥ or ( τ ( s ′ n,θ,ρ )) n ≥ . Then we call θ (resp. ρ ) the slope (resp.the intercept) of a .Applying so-called Subspace Theorem, Adamczewski and Bugeaud [3] established anew transcendence criterion for p -adic numbers. Theorem 1.1.
Let a = ( a n ) n ≥ be a non-ultimately periodic sequence over P . Set ξ := P ∞ n =0 a n p n ∈ Q p . If the sequence a is automatic, primitive morphic, or Sturmian, thenthe p -adic number ξ is transcendental. In this paper, we study p -adic numbers which satisfy the assumption of Theorem 1.1in more detail. For ξ ∈ Q p and an integer n ≥
1, we define w n ( ξ ) (resp. w ∗ n ( ξ )) to be thesupremum of the real number w (resp. w ∗ ) which satisfy0 < | P ( ξ ) | p ≤ H ( P ) − w − (resp. 0 < | ξ − α | p ≤ H ( α ) − w ∗ − )for infinitely many integer polynomials P ( X ) of degree at most n (resp. algebraic numbers α ∈ Q p of degree at most n ). Here, H ( P ), which is called the height of P ( X ), is definedby the maximum of the usual absolute values of the coefficients of P ( X ) and H ( α ), whichis called the height of α , is defined by the height of the minimal polynomial of α over Z .We set w ( ξ ) := lim sup n →∞ w n ( ξ ) n , w ∗ ( ξ ) := lim sup n →∞ w ∗ n ( ξ ) n . A p -adic number ξ is said to be an A -number if w ( ξ ) = 0; S -number if 0 < w ( ξ ) < + ∞ ; T -number if w ( ξ ) = + ∞ and w n ( ξ ) < + ∞ for all n ; U -number if w ( ξ ) = + ∞ and w n ( ξ ) = + ∞ for some n. Mahler [17] first introduced the classification. A p -adic number is algebraic if and onlyif it is an A -number. Almost all p -adic numbers are S -numbers in the sense of Haarmeasure. It is known that there exist uncountably many T -numbers. Liouville numbersare U -numbers, for example P ∞ n =1 p n ! . Replacing w n and w with w ∗ n and w ∗ , we define A ∗ -, S ∗ -, T ∗ -, and U ∗ -number as above. It is known that the two classification of p -adicnumbers coincide. Let n ≥ U -number (resp. a U ∗ -number) ξ ∈ Q p ,we say that ξ is a U n -number (resp. a U ∗ n -number ) if w n ( ξ ) is infinite and w m ( ξ ) are finite(resp. w ∗ n ( ξ ) is infinite and w ∗ m ( ξ ) are finite) for all 1 ≤ m < n . The detail is found in [9,Section 9.3].We now state the main results. Theorem 1.2.
Let a = ( a n ) n ≥ be a non-ultimately periodic sequence over P . Set ξ := P ∞ n =0 a n p n ∈ Q p . If the sequence a is automatic, primitive morphic, or Sturmian with itsslope whose continued fraction expansion has bounded partial quotients, then the p -adic AHLER’S CLASSIFICATION AND A CERTAIN CLASS OF p -ADIC NUMBERS 3 number ξ is an S - or T -number. Furthermore, if the sequence a is Strumian with itsslope whose continued fraction expansion has unbounded partial quotients, then the p -adicnumber ξ is a U -number. Theorem 1.2 is an extension of Theorem 1.1 and an analogue of Th´eor`emes 3.1, 4.2,and 5.1 in [5].
Theorem 1.3.
Let θ > be a real number whose continued fraction expansion has boundedpartial quotients, θ ′ > be a real number whose continued fraction expansion has un-bounded partial quotients, and ρ, ρ ′ be real numbers. Then the p -adic numbers ∞ X n =1 p ⌊ nθ + ρ ⌋ , ∞ X n =1 p ⌊ nθ ′ + ρ ′ ⌋ are algebraically independent. Theorem 1.3 is an analogue of Corollaire 3.2 in [5].This paper is organized as follows. In Section 2, we state Theorems 2.9 and 2.10, andprove the main results assuming Theorems 2.9 and 2.10. We prepare some lemmas toprove Theorems 2.9 and 2.10 in Section 3. In Section 4, we prove Theorems 2.9 and 2.10.2.
Extension of the main results
Let a = ( a n ) n ≥ be a sequence over a finite set A . The k -kernel of a = ( a n ) n ≥ is theset of all sequences ( a k i m + j ) m ≥ , where i ≥ ≤ j < k i .Eilenberg [14] characterized k -automatic sequences. Lemma 2.1.
Let k ≥ be an integer. Then a sequence is k -automatic if and only if its k -kernel is finite. We say that the sequence a = ( a n ) n ≥ is k -uniform morphic if there exist finite sets A , B , a k -uniform morphism σ : A ∗ → A ∗ which is prolongable on some a ∈ A , and acoding τ : A ∗ → B ∗ such that a = lim n →∞ τ ( σ n ( a )). Then we call A the initial alphabet associated with a .Cobham [12] showed another characterization of k -automatic sequences using k -uniformmorphic sequences. Lemma 2.2.
Let k ≥ be an integer. Then a sequence is k -automatic if and only if it is k -uniform morphic. The complexity function of the sequence a is given by p ( a , n ) := Card { a i a i +1 . . . a i + n − | i ≥ } , for n ≥ . Let ρ be a real number. We say that a satisfies Condition ( ∗ ) ρ if there exist sequencesof finite words ( U n ) n ≥ , ( V n ) n ≥ and a sequence of nonnegative real numbers ( w n ) n ≥ suchthat (i): the word U n V w n n is the prefix of a for all n ≥ (ii): | U n V w n n | / | U n V n | ≥ ρ for all n ≥ (iii): the sequence ( | V w n n | ) n ≥ is strictly increasing.The Diophantine exponent of a , first introduced in [2], is defined to be the supremum of areal number ρ such that a satisfy Condition ( ∗ ) ρ . We denote by Dio( a ) the Diophantineexponent of a . It is immediate that1 ≤ Dio( a ) ≤ + ∞ . TOMOHIRO OOTO
We recall known results about Diophantine exponents and complexity function forautomatic sequences, primitive morphic sequences, and Strumian sequences.Adamczewski and Cassaigne [1] estimated the Diophantine exponent of k -automaticsequences. Lemma 2.3.
Let k ≥ be an integer. Let a be a non-ultimately periodic and k -automaticsequence. Let m be a cardinality of the k -kernel of a . Then we have Dio( a ) < k m . Moss´e’s result [19] implies the following lemma.
Lemma 2.4.
Let a be a non-ultimately periodic and primitive morphic sequence. Thenthe Diophantine exponent of a is finite. Adamczewski and Bugeaud [5] established a relation between Strumian sequences andDiophantine exponents.
Lemma 2.5.
Let a be a Strumian sequence with slope θ . Then the continued fractionexpansion of θ has bounded partial quotients if and only if the Diophantine exponent of a is finite. It is known that automatic sequences, primitive morphic sequences, and Sturmian se-quences have low complexity.
Lemma 2.6.
Let k ≥ be an integer and a be a k -automatic sequence. Let d be acardinality of the internal alphabet associated with a . Then we have for all n ≥ p ( a , n ) ≤ kd n. Proof.
See [7, Theorem 10.3.1] or [12]. (cid:3)
Lemma 2.7.
Let a be a primitive morphic sequence over a finite set of cardinality of b ≥ . Let v be the width of a primitive morphism σ which generates the sequence a .Then we have for all n ≥ p ( a , n ) ≤ v b − b n. Proof.
See [7, Theorem 10.4.12]. (cid:3)
Lemma 2.8.
Let a be a Sturmian sequence. Then we have for all n ≥ p ( a , n ) = n + 1 . Proof.
See [7, Theorem 10.5.8]. (cid:3)
Theorem 2.9.
Let a = ( a n ) n ≥ be a non-ultimately periodic sequence over P . Set ξ := P ∞ n =0 a n p n ∈ Q p . Assume that there exist integers n ≥ and κ ≥ such that for all n ≥ n , p ( a , n ) ≤ κn. Then the p -adic number ξ is an S -, T -, or U -number. Theorem 2.9 is an analogue of Th´eor`eme 1.1 in [5]. There is a real continued fractionanalogue of Theorem 2.9 in [10, Theorem 3.2].
AHLER’S CLASSIFICATION AND A CERTAIN CLASS OF p -ADIC NUMBERS 5 Theorem 2.10.
Let a = ( a n ) n ≥ be a non-ultimately periodic sequence over P . Set ξ := P ∞ n =0 a n p n ∈ Q p . Assume that there exist integers n ≥ and κ ≥ such that forall n ≥ n , p ( a , n ) ≤ κn. Then the Diophantine exponent of a is finite if and only if ξ is not a U -number. Fur-thermore, if the Diophantine exponent of a is finite, then we have w ( ξ ) ≤ κ + 1) (2 κ + 1) Dio( a ) − . (1)There are various versions of Theorem 2.10: b -ary expansion for real numbers [5],continued fraction expansion for real numbers [10], formal power series over a finite field,and its continued fraction expansion [20]. Proof of Theorem 1.2 assuming Theorems 2.9 and 2.10.
Since the sequence a is automatic,primitive morphic, or Strumian, ξ is an S -, T -, or U -number by Lemmas 2.6, 2.7, 2.8and Theorem 2.9. It follows from Lemmas 2.3, 2.4, 2.5 and Theorem 2.10 that ξ is a U -number if a is Strumian with its slope whose continued fraction expansion has unboundedpartial quotients, and ξ is an S - or T -number otherwise. (cid:3) Let θ and ρ be real numbers. For an integer n ≥
1, we put t n := ( n = ⌊ kθ + ρ ⌋ for some integer k, ,t ′ n := ( n = ⌈ kθ + ρ ⌉ for some integer k, . We also put t θ,ρ := ( t n ) n ≥ and t ′ θ,ρ := ( t ′ n ) n ≥ . The lemma below is well-known result. Lemma 2.11.
Let θ > be an irrational real number and ρ be a real number. Then wehave t θ,ρ = s ′ /θ, − ( ρ +1) /θ and t ′ θ,ρ = s /θ, − ( ρ +1) /θ . Lemma 2.12 (Mahler [17]) . Let ξ, η be p -adic numbers. If ξ and η are algebraicallydependent, then ξ and η are in the same class.Proof of Theorem 1.3 assuming Theorems 2.9 and 2.10. Set ξ := P ∞ n =1 p ⌊ nθ + ρ ⌋ and η := P ∞ n =1 p ⌊ nθ ′ + ρ ′ ⌋ . By the definition, the digits of ξ and η are t θ,ρ and t θ ′ ,ρ ′ , respectively.It follows from Lemma 2.11 that t θ,ρ (resp. t θ ′ ,ρ ′ ) is Strumian with its slope whose con-tinued fraction expansion has bounded (resp. unbounded) partial quotients. Therefore, ξ is an S - or T -number and η is a U -number by Theorem 1.2. Hence, we see that ξ and η are algebraically independent by Lemma 2.12. (cid:3) Preliminaries
We recall several facts about the exponents w n and w ∗ n . Theorem 3.1.
Let n ≥ be an integer and ξ be in Q p . Then we have w ∗ n ( ξ ) ≤ w n ( ξ ) ≤ w ∗ n ( ξ ) + n − . Proof.
See [18]. (cid:3)
TOMOHIRO OOTO
Theorem 3.2.
Let n ≥ be an integer and ξ ∈ Q p be not algebraic of degree at most n .Then we have w n ( ξ ) ≥ n, w ∗ n ( ξ ) ≥ n + 12 . Furthermore, if n = 2 , then w ∗ ( ξ ) ≥ .Proof. See [17, 18]. (cid:3)
We recall Liouville inequality, that is, a non trivial lower bound of differences of twoalgebraic numbers.
Lemma 3.3.
Let α, β ∈ Q p be distinct algebraic numbers of degree m, n , respectively.Then we have | α − β | p ≥ ( m + 1) − n ( n + 1) − m H ( α ) n H ( β ) m . Proof.
See [21, Lemma 2.5]. (cid:3)
Applying Lemma 3.3, we give an estimate for the value of w . Lemma 3.4.
Let ξ be in Q p and c , c , c , θ, ρ, δ be positive numbers. Let ( β j ) j ≥ be asequence of positive integers with β j < β j +1 ≤ c β θj for all j ≥ . Assume that there existsa sequence of distinct terms ( α j ) j ≥ with α j ∈ Q such that for all j ≥ c β ρj ≤ | ξ − α j | p ≤ c β δj ,H ( α j ) ≤ c β j . Then we have δ ≤ w ( ξ ) ≤ (1 + ρ ) θδ − . Remark.
There are several versions of Lemma 3.4 as in [4, 5, 6, 8, 10, 11, 13, 16, 20, 22].
Proof.
Let α be a rational number with sufficiently large height. We define the integer j ≥ β j ≤ c (4 c c H ( α )) θ/δ < β j +1 . Firstly, we consider the case α = α j . By theassumption, we obtain | ξ − α | p ≥ c β − − ρj ≥ c − − ρ c (4 c c ) − (1+ ρ ) θ/δ H ( α ) − (1+ ρ ) θ/δ . Next, we consider the other case. Then, by the assumption, we have H ( α ) < (4 c c ) − ( c − β j +1 ) δ/θ ≤ (4 c c ) − β δj . Therefore, we obtain | α − α j | p ≥ (4 H ( α ) H ( α j )) − > c β − − δj by Lemma 3.3. Hence, it follows that | ξ − α | p = | α − α j | p ≥ (4 H ( α ) H ( α j )) − ≥ − − θ/δ c − c − θ/δ c − − θ/δ H ( α ) − − θ/δ . By Theorem 3.1, we have w ( ξ ) = w ∗ ( ξ ). Thus, we obtain δ ≤ w ( ξ ) ≤ max (cid:18) (1 + ρ ) θδ − , θδ (cid:19) = (1 + ρ ) θδ − . (cid:3) AHLER’S CLASSIFICATION AND A CERTAIN CLASS OF p -ADIC NUMBERS 7 We denote by M Q the set of all prime numbers and ∞ . We denote by | · | ∞ the usualabsolute value in Q . For x = ( x , . . . , x n ) ∈ Q n and v ∈ M Q , we define the norm and the height of x by | x | v = max ≤ i ≤ n | x i | v and H ( x ) = Q v ∈ M Q | x | v .The proof of Theorem 2.9 mainly depends on the following theorem which is so-calledQuantitative Subspace Theorem and consequence of Corollary 3.2 in [15]. Theorem 3.5.
Let α ∈ Q p be an algebraic number of degree d and < ε < . Definelinear forms L ∞ ( X, Y, Z ) =
X, L ∞ ( X, Y, Z ) =
Y, L ∞ ( X, Y, Z ) =
Z,L p ( X, Y, Z ) =
X, L p ( X, Y, Z ) =
Y, L p ( X, Y, Z ) = αX − αY − Z. Then all integer solutions x = ( x , x , x ) of Y v ∈{ p, ∞} Y i =1 | L iv ( x ) | v ≤ | x | − ε ∞ with H ( x ) ≥ max (cid:18)(cid:16) √ d + 1 H ( α ) (cid:17) / d , /ε (cid:19) lie in the union of at most ε − log(3 ε − d ) log( ε − log 3 d ) proper linear subspaces of Q . Consider a vector hyperplane of Q n H = { ( x , . . . , x n ) ∈ Q n | y x + · · · + y n x n = 0 } , where y = ( y , . . . , y n ) ∈ Z n \ { } , gcd( y , . . . , y n ) = 1. The height of H , denoted by H ( H ), is defined to be | y | ∞ .The lemma below is easily seen. Lemma 3.6.
Let m, n be integers with ≤ m < n and x , . . . , x m ∈ Z n be linearlyindependent vectors such that | x | ∞ ≤ . . . ≤ | x m | ∞ . Then there exists a vector hyperplane H of Q n such that x , . . . , x m ∈ H and H ( H ) ≤ m ! | x m | m ∞ . Lemma 3.7.
Let U ∈ P ∗ , V ∈ P + , and r, s be lengths of the words U, V , respectively.Put ( a n ) n ≥ := U V and α := P ∞ n =0 a n p n ∈ Q p . Then we have H ( α ) ≤ p r + s .Proof. A straightforward computation shows that α = r − X n =0 a n p n + s − X m =0 a m + r p m + r ! ∞ X k =0 p ks ! = ( p s − P r − n =0 a n p n − P s − m =0 a m + r p m + r p s − . Therefore, we have H ( α ) ≤ max p s − , ( p s − r − X n =0 a n p n , s − X m =0 a m + r p m + r ! ≤ p r + s . (cid:3) TOMOHIRO OOTO
In order to prove Theorem 2.10, we show the following lemma.
Lemma 3.8.
Let a = ( a n ) n ≥ be a non-ultimately periodic sequence over P . Set ξ := P ∞ n =0 a n p n ∈ Q p . Then we have w ( ξ ) ≥ max(1 , Dio( a ) − . (2) Proof.
Since ξ is irrational, we have w ( ξ ) ≥ a ) >
1. Take a real number δ such that 1 < δ < Dio( a ). For n ≥
1, there exist finite words U n , V n and a positive rational number w n such that U n V w n n are the prefix of a , the sequence ( | V w n n | ) n ≥ is strictly increasing, and | U n V w n n | ≥ δ | U n V n | .For n ≥
1, we set rational number α n := ∞ X i =0 b ( n ) i p i where ( b ( n ) i ) i ≥ is the infinite word U n V n . Since ξ and α n have the same first | U n V w n n | -thdigits, we obtain | ξ − α n | ≤ p − δ | U n V n | ≤ H ( α n ) − δ by Lemma 3.7. Hence, we have (2). (cid:3) The following lemma is a slight improvement of a part of Lemma 9.1 in [10].
Lemma 3.9.
Let a = ( a n ) n ≥ be a sequence on a finite set A . Assume that there existintegers κ ≥ and n ≥ such that for all n ≥ n , p ( a , n ) ≤ κn. Then, for each n ≥ n , there exist finite words U n , V n over A and a positive rationalnumber w n such that the following hold: (i): U n V w n n is a prefix of a , (ii): | U n | ≤ κ | V n | , (iii): n/ ≤ | V n | ≤ κn , (iv): if U n is not an empty word, then the last letter of U n and V n are different, (v): | U n V w n n | / | U n V n | ≥ / (4 κ + 2) , (vi): | U n V n | ≤ ( κ + 1) n − ,Proof. For n ≥
1, we denote by A ( n ) the prefix of a of length n . By Pigeonhole principle,for each n ≥ n , there exists a finite word W n of length n such that the word appears to A (( κ +1) n ) at least twice. Therefore, for each n ≥ n , there exist finite words B n , D n , E n ∈A ∗ and C n ∈ A + such that A (( κ + 1) n ) = B n W n D n E n = B n C n W n E n . We take these words in such way that if B n is not empty, then the last letter of B n isdifferent from that of C n .We first consider the case of | C n | ≥ | W n | . Then, there exists F n ∈ A ∗ such that A (( κ + 1) n ) = B n W n F n W n E n . Put U n := B n , V n := W n F n , and w n := | W n F n W n | / | W n F n | . Since U n V w n n = B n W n F n W n ,the word U n V w n n is a prefix of a . It is obvious that | U n | ≤ ( κ − | V n | and n ≤ | V n | ≤ κn .By the definition, we have (iv) and (vi). Furthermore, we see that | U n V w n n || U n V n | = 1 + n | U n V n | ≥ κ . AHLER’S CLASSIFICATION AND A CERTAIN CLASS OF p -ADIC NUMBERS 9 We next consider the case of | C n | < | W n | . Since the two occurrences of W n do overlap,there exists a rational number d n > W n = C d n n . Put U n := B n , V n := C ⌈ d n / ⌉ n ,and w n := ( d n + 1) / ⌈ d n / ⌉ . Obviously, we have (i) and (iv). Since ⌈ d n / ⌉ ≤ d n and d n | C n | ≤ ⌈ d n / ⌉| C n | , we get n/ ≤ | V n | ≤ n . Using (iii) and | U n | ≤ κn −
1, we see (ii)and (vi). It is immediate that w n ≥ /
2. Hence, we obtain | U n V w n n || U n V n | = 1 + ⌈ ( w n − | V n |⌉| U n V n | ≥ w n − | U n | / | V n | + 1 ≥ / κ + 1 = 1 + 14 κ + 2 . (cid:3) Proof of Theorems 2.9 and 2.10
Proof of Theorem 2.9.
By Theorem 1B in [3], ξ is transcendental, that is, ξ is not an A -number. Therefore, it is sufficient to prove that if ξ is not a U -number, then ξ isnot a U -number. For n ≥ n , we take finite words U n , V n over P and positive rationalnumbers w n satisfying Lemma 3.9 (i)-(vi). We define a positive integer sequence ( n k ) k ≥ by n k +1 = 4( κ + 1) n k for k ≥
0. We set r k := | U n k | , s k := | V n k | , and t k := | U n k V n k | for k ≥
0. Then a straightforward computation shows that 2 t k ≤ t k +1 ≤ ct k , r k ≤ κs k for k ≥
0, and ( s k ) k ≥ is strictly increasing, where c = 8( κ + 1) . For k ≥
0, there exists aninteger p k such that p k p s k − ∞ X i =0 b ( k ) i p i where ( b ( k ) i ) i ≥ is the infinite word U n k V n k . Since ξ and p k / ( p s k −
1) have the same first | U n k V w nk n k | -th digits, we obtain (cid:12)(cid:12)(cid:12)(cid:12) ξ − p k p s k − (cid:12)(cid:12)(cid:12)(cid:12) p ≤ p − wt k , where w = 1+1 / (4 κ +2). Since the sequence ( s k ) k ≥ is strictly increasing, we may assumethat t ≥ α ∈ Q p be an algebraic number of degree d ≥ H ( α ) ≥ max( d +1 , p s , κ +2 ).We define an integer j ≥ p s j − ≤ H ( α ) < p s j and a real number χ by | ξ − α | p = H ( α ) − χ . Without loss of generality, we may assume that χ >
0. Put M := max { m ∈ Z | p wc m − t j < H ( α ) χ } . In what follows, we estimate an upper bound of M . Therefore,we may assume that M ≥
1. Then we obtain p wt j + h ≤ p wc M − t j for all 0 ≤ h ≤ M − | p s j + h α − α − p j + h | p = (cid:12)(cid:12)(cid:12)(cid:12) α − p j + h p s j + h − (cid:12)(cid:12)(cid:12)(cid:12) p ≤ max (cid:12)(cid:12)(cid:12)(cid:12) ξ − p j + h p s j + h − (cid:12)(cid:12)(cid:12)(cid:12) p , | ξ − α | p ! ≤ p − wt j + h for 0 ≤ h ≤ M −
1. We define linear forms by L ∞ ( X, Y, Z ) =
X, L ∞ ( X, Y, Z ) =
Y, L ∞ ( X, Y, Z ) =
Z,L p ( X, Y, Z ) =
X, L p ( X, Y, Z ) =
Y, L p ( X, Y, Z ) = αX − αY − Z, and put x h := ( p s j + h , , p j + h ) for 0 ≤ h ≤ M −
1. By the proof of Lemma 3.7, we obtain Y v ∈{ p, ∞} Y i =1 | L iv ( x h ) | v ≤ | x h | − / (4 κ +2) ∞ for all 0 ≤ h ≤ M −
1. We also have H ( x h ) = | x h | ∞ ≥ p s j + h ≥ H ( α ) ≥ max (cid:18)(cid:16) √ d + 1 H ( α ) (cid:17) / d , κ +2 (cid:19) for all 0 ≤ h ≤ M −
1. Hence, by Theorem 3.5, for all 0 ≤ h ≤ M −
1, we obtain x h inthe union of N proper linear subspaces of Q , where N = ⌊ (2 κ + 1) log(6(2 κ +1) d ) log(2(2 κ + 1) log 3 d ) ⌋ .Assume that one of these linear subspaces of Q contains L + 1 points of the set { x h | ≤ h ≤ M − } , where L = ⌈ log ((2 κ + 1)(4 d + 6 + log p (2 d +1 ( d + 1)))) ⌉ . It followsthat there exist ( x, y, z ) ∈ Z \ { } such that xp s j + ik + y + zp j + i k = 0 , (0 ≤ k ≤ L ) , where 0 ≤ i < i < . . . < i L < M . Since x i and x i are linearly independent, we chose( x, y, z ) ∈ Z \ { } such that max( | x | , | y | , | z | ) ≤ p t j + i by Lemma 3.6. Since ( s k ) k ≥ isstrictly increasing, we have z = 0. A straightforward computation shows that(1 − p s j + ik ) α = p s j + ik xz + yz − ( p s j + ik α − α − p j + i k ) ,α − yz = p s j + ik α + p s j + ik xz − ( p s j + ik α − α − p j + i k )for all 0 ≤ k ≤ L . Therefore, we obtain | α | p = | (1 − p s j + ik ) α | p ≤ max (cid:18) p − s j + ik (cid:12)(cid:12)(cid:12) xz (cid:12)(cid:12)(cid:12) p , (cid:12)(cid:12)(cid:12) yz (cid:12)(cid:12)(cid:12) p , p − wt j + ik (cid:19) ≤ max( | z | , p − wt j + ik ) ≤ p t j + i . Hence, we have (cid:12)(cid:12)(cid:12) α − yz (cid:12)(cid:12)(cid:12) p ≤ max(2 p t j + i − s j + iL , | z | p − s j + iL , p − wt j + iL ) = 2 p t j + i − s j + iL . (3)It follows from Lemma 3.3 that (cid:12)(cid:12)(cid:12) α − yz (cid:12)(cid:12)(cid:12) p ≥ − d ( d + 1) − H ( α ) − H (cid:16) yz (cid:17) − d ≥ − d ( d + 1) − p − dt j + i − s j . (4)By the properties of ( s k ) k ≥ and ( t k ) k ≥ , we have s j + i L ≥ t j + i L κ + 1 = 12 κ + 1 t j + i L t j + i L − · · · t j + i t j + i t j + i ≥ L t j + i κ + 2 . (5)Applying (3), (4), and (5), we obtain t j + i ≤ (4 κ + 2) log p (2 d +1 ( d + 1))2 L − (4 κ + 2)(2 d + 3) ≤ , which is contradiction.Hence, we get M ≤ LN . By the definition of M , we have H ( α ) χ ≤ p wc M t j ≤ p wc M +1 t j − ≤ p wc M +1 (2 κ +1) s j − ≤ H ( α ) wc M +1 (2 κ +1) . AHLER’S CLASSIFICATION AND A CERTAIN CLASS OF p -ADIC NUMBERS 11 Therefore, we obtain | ξ − α | p ≥ H ( α ) − wc LN +1 (2 κ +1) , which implies w ∗ d ( ξ ) ≤ max( w ( ξ ) , wc LN +1 (2 κ + 1)) . This completes the proof. (cid:3)
Proof of Theorem 2.10.
We first assume that ξ is not a U -number, that is, w ( ξ ) is finite.Then Dio( a ) is finite by Lemma 3.8.We next assume that Dio( a ) is finite. For n ≥ n , take finite words U n , V n and a rationalnumber w n satisfying Lemma 3.9 (i)-(vi). For n ≥ n , we set rational numbers α n := ∞ X i =0 b ( n ) i p i where ( b ( n ) i ) i ≥ is the infinite word U n V n . Since ξ and α n have the same first | U n V w n n | -thdigits, we obtain | ξ − α n | ≤ p − ( κ +2 ) | U n V n | . Take a real number δ which is greater than Dio( a ). Note that δ >
1. By the definition ofthe Diophantine exponent, there exists an integer n ≥ n such that for all n ≥ n | ξ − α n | ≥ p − δ | U n V n | . We define a positive integer sequence ( n k ) k ≥ by n k +1 = 2( κ + 1) n k for k ≥
1. Set β k := p | U nk V nk | for k ≥
1. It follows from Lemma 3.9 (iii),(vi) that for n ≥ β k < β k +1 ≤ β κ +1) k . By Lemma 3.7, we have H ( α n k ) ≤ β k for k ≥
1. Hence, we obtain (1) by Lemma 3.4. (cid:3)
Acknowledgements.
I would like to thank Prof. Shigeki Akiyama for improving thelanguage and structure of this paper. I wish to thank the referee for a careful reading andseveral helpful comments.
References [1] B. Adamczewski, J. Cassaigne,
Diophantine properties of real numbers generated by finite automata ,Compos. Math. (2006), no. 6, 1351-1372.[2] B. Adamczewski, Y. Bugeaud,
Dynamics for β -shifts and Diophantine approximation , Ergodic TheoryDynam. Systems (2007), no. 6, 1695-1711.[3] B. Adamczewski, Y. Bugeaud, On the complexity of algebraic numbers. I. Expansions in integer bases ,Ann. of Math. (2) (2007), no. 2, 547-565.[4] B. Adamczewski, Y. Bugeaud,
Mesures de transcendance et aspects quantitatifs de la m´ethode deThue-Siegel-Roth-Schmidt , (French) Proc. Lond. Math. Soc. (3) (2010), no. 1, 1-26.[5] B. Adamczewski, Y. Bugeaud,
Nombres r´eels de complexit´e sous-lin´eaire: mesures d’irrationalit´e etde transcendance , (French) J. Reine Angew. Math. (2011), 65-98.[6] K. Alladi, M. L. Robinson,
Legendre polynomials and irrationality , J. Reine Angew. Math. (1980),137-155.[7] J.-P. Allouche, J. Shallit,
Automatic Sequences: Theory, Applications, Generalizations , CambridgeUniversity Press, Cambridge (2003).[8] M. Amou,
Approximation to certain transcendental decimal fractions by algebraic numbers , J. NumberTheory (1991), no. 2, 231-241.[9] Y. Bugeaud, Approximation by algebraic numbers , Cambridge Tracts in Mathematics,
CambridgeUniversity Press, Cambridge, 2004. [10] Y. Bugeaud,
Continued fractions with low complexity: transcendence measures and quadratic ap-proximation , Compos. Math. (2012), no. 3, 718-750.[11] Y. Bugeaud, T. Pejkovi´c,
Quadratic approximation in Q p , Int. J. Number Theory (2015), no. 1,193-209.[12] A. Cobham, Uniform tag sequences , Math. Systems Theory (1972), 164-192.[13] L. V. Danilov, Rational approximations of some functions at rational points , (Russian) Mat. Zametki (1978), no. 4, 449-458, 589.[14] S. Eilenberg, Automata, languages, and machines, Vol. A , Pure and Applied Mathematics, Vol. Academic Press, New York, 1974.[15] J. H. Evertse, R. G. Ferretti,
A further improvement of the quantitative subspace theorem , Ann. ofMath. (2) (2013), no. 2, 513-590.[16] A. Firicel,
Rational approximations to algebraic Laurent series with coefficients in a finite field , ActaArith. (2013), no. 4, 297-322.[17] K. Mahler, ¨Uber eine Klasseneinteilung der p -adischen Zahlen , (German) Mathematica Leiden (1934/35), 177-185.[18] J. F. Morrison, Approximation of p -adic numbers by algebraic numbers of bounded degree , J. NumberTheory (1978), no. 3, 334-350.[19] B. Moss´e, Puissances de mots et reconnaissabilit´e des points fixes d’une substitution , (French) The-oret. Comput. Sci. (1992), no. 2, 327-334.[20] T. Ooto, Quadratic approximation in F q (( T − )), Preprint arXiv 1512.04041.[21] T. Pejkovi´c, Polynomial root separation and applications , PhD Thesis, Universit´e de Strasbourg andUniversity of Zagreb, Strasbourg, 2012.[22] J. F. Voloch,
Diophantine approximation in finite characteristic , Period. Math. Hungar. (1988),217-225. Graduate School of Pure and Applied Sciences, University of Tsukuba, Tennodai 1-1-1, Tsukuba, Ibaraki, 305-8571, Japan
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