Arithmetic and Metric Properties of p-adic Alternating Engel Series Expansions

Transcription

1 International Journal of Algebra, Vol 2, 2008, no 8, Arithmetic and Metric Proerties of -Adic Alternating Engel Series Exansions Yue-Hua Liu and Lu-Ming Shen Science College of Hunan Agriculture University Changsha, Hunan, 40082, PR China Abstract In this aer, the alternating Engel exansion over the field of -adic series is studied Metric roerties, such as strong and weak number laws, central it theorem, and iterated logarithm law, of the digits occurring in this exansion are considered At the same time, the aroximation orders by rational fractions which are the artial sums of the series are investigated Mathematics Subject Classification: Primary K55, T06; Secondly A7 Keywords: alternating Engel exansion; -adic series; metric roerty; convergence seed Introduction Let Q be the field of rational numbers, be a rime number and Q the comletion of Q with resect to the adic absolute value defined on Q by([6]) 0 0 and A a if A a r, where is not divisible with r and s s The exonent a in this definition is the adic valuation of A, which denoted by ν (A) Then Q is the field of adic numbers with valuation, the extension of the original valuation on Q, which has the roerties A 0 with A 0 iff A 0, AB A B, and A + B max( A, B ) with equality when A B Corresonding author lum s@26com

2 384 Yue-Hua Liu and Lu-Ming Shen It is familiar that the above non-archimedean valuation leads to an ultrametric distance function ρ on Q, with ρ(a, B) A B, making Q into a comlete metric sace with resect to ρ As is common, we define the order ν (A) ofa by A ν(a), and set ν (0) + Then ν (AB) ν (A) + ν (B), ν ( A B )ν (A) ν (B) ifb 0, and ν (A + B) min(ν (A),ν (B)) with equality when ν (A) ν (B) It is well known that every A Q has a unique series reresentation([6]) A nν (A) c n n, c n {0,,, } In the discussion below we call A ν c (A) n 0 n n the fractional art of A Then A S, where we define S { A : A Q } Q The set S is not multilicatively or additively closed The function A and the set S have been used in the study of certain tyes of -adic continued fractions by Laohakosol([7]) in articular The fraction art A was used by AKnofmacher and JKnofmacher([,2]) to derive some new unique series exansions for any element A Q, including articular of certain Sylvester, Engel and Lüroth exansions of arbitrary real numbers into series with rational terms([5], Cha IV) In the corresonding case of adic Lüroth tye exansions ergodic and other metric roerties have been investigated by A and JKnofmacher[3] For both the -adic continued fractions and Lüroth exansions, ergodicity of the corresonding transformations were used to derive the results However, in the case of Engel, Peter J Grabner and Arnold Knofmacher have obtained some similar metric asymtotic results for the -adic Lüroth tye exansions For Oenheim exansions over the field of Laurent series, Ai-hua Fan and Jun Wu([5]) studied the metric roerties and the aroximation orders by rational fractions which are the artial sums of these series The aim of this aer is to derive metric roerties of the digits {a n (x)} for the alternating Engel exansion exansions of -adic series Let I denote the valuation ideal Z in the ring of -adic integers Z and let P denote robability with resect to the Haar measure on (Q, +) normalized by P(I) The Haar measure on I is the roduct measure on {0,, } N defined by P({x}) for each factor and any element x {0,, } Considering {a n (x)} n as a sequence of random variables defined on the robability sace (I,P), we find the finite dimensional joint distributions of this sequence It turns out that both {a n (x)} and ν (a n (x)) are Markov chains and that their transition robabilities are by exlicit formulas These allow us to rove that { n (x) ν (a n (x)) ν (a n+ (x))} n 0 is a sequence of iid geomitrical random variable As a consequence of the main result and the classical law of large numbers and central it theorem, we deduce the metric

3 -Adic alternating Engel series exansions 385 roerties of n j0 j and the aroximation order of a tyical -adic series by rational functions 2 Metric theory We are now in the osition to introduce the algorithm of alternating Engel exansion exansion over the field of -adic series Given any A Q, note that if A a 0 S, then ν (A a 0 ) Then define A A a 0 Suose A n (n ) is defined If A n 0, then let a n A n, and define A n+ A n a n If A n 0, then this recursive stos We call {a n } the digits of A Lemma 2 If A n 0, then ν(a n+ ) ν (a n ) Theorem 22 Every x Q has a finite or an infinite convergent (relative to ρ) exansion of the form x a 0 + a + + n2 ( ) n+ a a n () where a n S, a 0 A, and ν (a ), ν (a n+ ) ν (a n ) Furthermore, the exansion is unique The roofs of the Lemma2 and the Theorem22 are very similar to that of Oenheim series of -adic([2]), we will omit the rocess In this section, we are concentrated in deriving the metric roerties of digits occurring in the exansion Actually, we rove that the sequence of digits {a n (x)} n is a Markov chain and so is {ν (a n (x))} n We also obtain the finite dimensional joint distributions for both Markov chains Most imortantly, we find for any n, 0 (x) : ν (a (x)), n (x) :ν (a n (x)) ν (a n+ (x)) are indeendent and identically distributed as a sequence of random variables Definition 23 A finite sequence {k,,k n } S is said to be admissible, if it satisfies the admissibility condition ν (a n+ ) 2ν (a n ) Lemma 24 Let x I whose alternating Engel exansion exansion of -adic series is x n c n (x), c n (x) ( ) n+ a (x) a n (x) We have c n+ (x) c n (x) for all n

4 386 Yue-Hua Liu and Lu-Ming Shen ProofNotice that ν (c n (x)) n i ν (a i (x)) The difference ν (c n+ (x)) ν (c n (x)) is equal to ν (a n+ (x)), which is strictly ositive by the admissible condition Lemma 25 Suose {a,a 2,,a n } S n is admissible, then {x I : a (x) a,,a n (x) a n } : B(a,,a n ) called n-digits cylinder is equal to the disc B(C n,d n ) with center and diameter C n : a + j2 ( ) j a a n D n P n j ν(a j)+2ν (a n) (2) Proof For any x B(a,,a n ), by Lemma 2, x C n c n+ P n j ν(a j)+ν (a n+ (x)) P n j ν(a j)+2ν (a n) It follow that x B(C n,b n ) Thus we get {x I : a (x) a,,a n (x) a n } B(C n,b n ) Conversely, for any y B(C n,d n ), then y j c j(y) We are going to show that a j (y) a j for all j n Firstly, we rove a (x) a by contradiction Suose a (y) a There are two cases Case I: ν (a (y) ν (a ) In this case, we have By Lemma22, we have a (y) max( a a (y), ) a a (y) c j(y) (j 2); > c j, (2 j n) a Therefore max( a (y), ) > max( max a max( c j(y), c j ), su c j (y) ) 2 j n j n+ It follows that y C n a (y) max( a a (y), ) ν(a ) a a

5 -Adic alternating Engel series exansions 387 By the admissibility condition, we may check that log D m is decreasing, so that D n D 2ν(a ) < ν(a ) y C n, which contradicts the fact y B(C n,d n ) Case II: ν (a (y)) ν (a ) In this case, using (4), we can rove by induction on j that a (y) 2 c j (y), (j 2; ) a 2 c j (y), (2 j n) Thus Hence a (y) 2 a a a (y) 2 > max( max max( c j(y), c j ), su c j (y) ) 2 j n j n+ y C n a (y) 2 2ν(a) >D D n a a Hence we have roved a (y) a j n In the same way, a j (x) a j, for all Proosition 26 Suose {a,,,a n,a n+ } S is admissible, then P{B(a,,a n )} P n j ν(a j)+2ν (a n) (3) Proof By Lemma25, {x I : a (x) a,,a n (x) a n } is the disc B(C n,d n ), thus by the definition of robability P, we have P{B(C n,d n )} D n P n j ν(a j)+2ν (a n) Proosition 27 Suose that {x I 0 : a (x) a,,a n (x) a n } is an admissible sequence, then we have P{a n+ (x) a n+ a n (x) a n } Proof By Proosition26, an a n+ 2 P{a n+ (x) a n+ a n (x) a n,,a (x) a } P{a (x) a,,a n+ (x) a n+ } P{a (x) a,,a n (x) a n } 2ν(a n+) a n ν(an) a n+ 2

6 388 Yue-Hua Liu and Lu-Ming Shen On the other hand, P{a n+ (x) a n+ a n (x) a n } P{a n (x) a n,a n+ (x) a n+ } P{a n (x) a n } P{aj (x) l j, j n,a n (x) a n,a n+ (x) a n+ } P{aj (x) l j, j n,a n (x) a n } P n j ν(a j)+2ν (a n+ ) P n j ν(a j)+2ν (a n) a n a n+ 2 where both the summation in the numerators are taken over all sequences {l,,l n } S, such that {l,,l n,k n } is admissible Consequently, the sequence {a j (x)} j forms a Markov chain with the transition robability P{a n+ (x) a n+ a n (x) a n } a n a n+ 2 Proosition 28 Suose that {k,,k n } is an admissible sequence, then we have P{x I : ν (a (x)) ν (k ),,ν (a n (x)) ν (k n )} ( ) n ν(kn), and consequently, the sequence {ν (a j (x))} j forms a Markov chain with the transition robability a n P{ν (a n+ (x)) ν (a n+ ) ν (a n (x)) ν (a n )} ( ) a n+ ProofFirst remark that for any n and any a, l S such that ν (k) ν (l), we have ν (k) ν (l), by Proosition26, we have P{x I : ν (a (x)) ν (k ),,ν (a n (x)) ν (k n )} P{a (x) l,,a n (x) l n } P n j ν(l j)+2ν (l n) ( ) n ν(kn) where both the summation in the numerators are taken over all sequences {l,,l n } S, such that ν (l j )ν (k j ), j n It is trivial that P{ν (a n+ (x)) ν (a n+ ) ν (a n (x)) ν (a n )} ( ) a n 2 a n+

7 -Adic alternating Engel series exansions 389 Theorem 29 { n } n 0 is a sequence of indeendent and identically distributed random variables Furthermore, deg a (x) is geometrical, ie for any m, P{x I : (x) m} (4) m Proof P{x I : n+ m} P(B(a,,a n+ )) a,,a n+ P(B(a,,a n )) 2ν(an+) a,,a n a n+ P(B(d,,a n ))( ) m m a,,a n where the summation on {a,,a n,a n+ } is over all the admissible sequence with 2ν (a n (x)) ν (a n+ (x) m At the same time, it is easy to see, for any m, P{x I : (x) m} m For any ositive integer m,m 2,,m n+, P{x I : j (x) m j, j n +} P{x I : a (x) a,,a n+ (x) a n+ } a,,a n+ P(a,,a n ) 2ν(an+) 2ν(an) a,,a n a n+ P(a,,a n ) m n+ a,,a n P{x : j (x) m j, j n} m n+ n P{ j (x) m j+ } j0 where all the summations are taken over all admissible sequence with 2ν (a j ) ν (a j )m j, for all j n + By this theorem, it can be easily testified the following roosition Proosition 20 E( n (x)), Var( n(x)) ( ) 2 (5)

8 390 Yue-Hua Liu and Lu-Ming Shen Proof E( n (x)) lp( n (x) l) ( ) l l l l, E( 2 n (x)) l 2 P( n (x) l) l +2 ( ), 2 from which the formula for Var( n (x)) is immediateas consequence of Theorem 29, and the classical it theorems on iid random variables, we get immediately the following metric roerties: Theorem 2 For the alternating Engel exansion exansions over the field of -adic series, we have: + (i) For P-almost all x I, n (x) > log φ(n) io iff φ(n) + n ν (a j (x)) ν (a n+ (x)) (ii) P{x I : j <t} t e u2 /2 du n/( ) 2π (iii) For P-almost all x I, n ( ν (a j (x)) ν (a n+ (x))) j (iv) For P-almost all x I 0, (6) su ν (a j (x)) ν (a n+ (x)) j n 2n log log n (7) and inf ν (a j (x)) ν (a n+ (x)) j 2n log log n n (8) An direct alication of (i) yields: Corollary 22 For P-almost all x I, su 2ν (a n (x)) ν (a n+ (x)) log n (9) log log n

9 -Adic alternating Engel series exansions 39 Theorem 23 j (x) converges in robability to That is n log n j to say, for any fixed ɛ>0, P{x I : j (x) ( ) >ɛ} 0 (0) n log n j Proof Fix n For any k n, define { } k U k (x) (x), if k(x) n log n, 0 0, otherwise V k(x) k (x) P{x I : k(x) ( ) >ɛ} n log n k n k P{x I : U k(x) ( ) >ɛ} + P{x I : V k (x) 0} n log n k : P(A n )+P(B n ) Since P(B n ) P{x I : V k (x) 0} P{x I : k(x) > log n} k k l: l >n log n Notice that E(U k (x)) l: l n log n k (x) n log n k O( l log n ) k(x) dp l: l n log n l P( k (x) l) l l ( ) log n () Var(U k (x)) E(U 2 k (x)) l: l n log n Then by Chebyshev s inequality, we get P{x I 0 : U k (x) E(U k (x)) >ɛ k k ( ) l cn log n E(U k (x))} k Var( U k (x)) k n 2 log n O( n (ɛ E(U k (x))) 2 2 log 2 n )O( log n ) k

10 392 Yue-Hua Liu and Lu-Ming Shen That is to say, n k E(U U k (x) converges in robability to Since k(x)) k E(U k ) ( ) log n as n +, we get n log n Thus P(A n ) 0, as n + E(U k (x)) k 3 The seed of convergence In this section, we consider the aroximation of -adic series by its convergent For any x I, we define its artial sum C n (x) a (x) + a n (x) Theorem 3 For alternating Engel exansion exansions of -adic series, n (log x C n (x) j2 ν (a j (x))) (2) j ProofAs we can see, C n+ (x) C n (x), then log x C n (x) ν (a n+ (x)) ν (a j (x)) j (x)+ν (a (x)), j j then by theorem2, for P-almost all x I, n (log x C n (x) Theorem 32 For P-almost all x I 0, where G(x) + n 2 n n(x) ν (a j (x))) j 2 n log x C n G(x)

11 -Adic alternating Engel series exansions 393 ProofSince 0 (x) ν (a (x)), n (x) 2ν (a n (x)) ν (a n+ (x)), n, By Theorem29, Theorem2 and Toelitz Lemma, for P-almost all x I 0, Notice that and j0 2 (log ν (a n+ (x)) n x C n (x) 2 n+ 2 n ν (a n+ (x)) 2 n j0 j (x) 2 j k (x) (2 L) dp j (2 L) j < +, j0 Hence by the Beo-levi theorem, G(x) exists, for P-almost all x I References [] AKnofmacher and JKnofmacher Metric roerties of some secial -adic series exansions Acta Arith, 76(996), -9 [2] AKnofmacher and JKnofmacher Infinite series exansions for -adic numbers JNumber Theory, 4(992), 3-45 [3] A Knofmacher and J Knofmacher Inverse olynomial exansions of Laurent series Constr Arox, 4(988), [4] Peter J Grabner, Arnold Knofmacher Arithmetic and Metric Proerties of -adic Engel Series Exansions Publicationes Mathematicae Debrecen, 63(2003), [5] Ai-Hua Fan and Jun Wu Aroximation orders of formal Laurent series by Oenheim rational functions, J Arox Theory 2(2003), [6] NKoblitz -Adic Numbers, -adic Analysis, and Zeta-Functions Sringer-Verlag, New York,984 [7] VLaohakosol A characterization of rational numbers by -adic Ruban continued fractions J Austral math Soc Ser A, 39(985), Received: October 2, 2007