PETER J. GRABNER AND ARNOLD KNOPFMACHER

ARITHMETIC AND METRIC PROPERTIES OF -ADIC ENGEL SERIES EXPANSIONS PETER J. GRABNER AND ARNOLD KNOPFMACHER Abstract. We derive a characterization of rational numbers in terms of their unique -adic Engel series exansions. Thereafter we investigate metric roerties for the rational digits occurring in these -adic Engel exansions. In articular, we obtain limiting distributions for the -adic order of the digits and the -adic order of aroximation by the artial sums of the series exansions.. Introduction Let Q be the field of rational numbers, a rime number and Q the comletion of Q with resect to the -adic absolute value defined on Q by (cf. Koblitz [8 or Schikhof [2) (.) 0 = 0 and A = a if A = ar, where rs. s The exonent a in this definition is the -adic valuation of A, which we denote by v (A). It is well known that every A Q has a unique series reresentation A = c n n, c n {0,,2,..., }. n=v (A) In the discussion below we call the finite sum A = v (A) n 0 c n n the fractional art of A. Then A S, where we define S = { A : A Q } Q. This set S is neither multilicatively nor additively closed. Recently the fractional art A was used by A. Knofmacher and J. Knofmacher [5, 6, to derive some new unique series exansions for any element A Q, including in articular analogues of certain Sylvester, Engel and Lüroth exansions of arbitrary real numbers into series with rational terms (cf. [0, Cha. IV). In the corresonding case of -adic Lüroth tye exansions ergodic and other metric roerties have recently been investigated by A. and J. Knofmacher [7. 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 exansions the underlying transformation is not ergodic. The growth conditions satisfied by the digits Date: October 5, 2002. 99 Mathematics Subject Classification. Primary: K55; Secondary:K4. Key words and hrases. Engel series, -adic numbers, metric roerties. This author is suorted by the START roject Y96-MAT of the Austrian Science Fund.

2 P. J. GRABNER AND A. KNOPFMACHER suggest that an aroach via Markov chains could be used. A similar aroach was used to study metric roerties of Engel exansions over the field of formal Laurent series over a finite field in [4. Given A Q, now note that A = a 0 S iff v (A ) where A = A a 0. As in [5, if A n 0(n > 0) is already defined, we then let (.2) a n = and ut A n+ = a n A n. A n If some A m = 0, this recursive rocess stos. It was shown in [5 that this algorithm leads to a finite or convergent (relative to v ) Engel-tye series exansion (.3) A = a 0 + a + r 2 a a r, where a r S, a 0 = A, and v (a r+ ) v (a r ) for r. Furthermore this exansion is unique for A subject to the receding conditions on the digits a r. For notational convenience we set n = a 0 + r= a a r, where = a...a n. In [6 it was claimed (see Proosition 4.2) that rational numbers are characterized by finite -adic Engel exansions. However, the following examle of an infinite -adic Engel exansion for a rational number shows that this result is incorrect. For convenience in the sequel we denote the -adic Engel exansion + + + by (a,a 2,a 3,...). a a a 2 a a 2 a 3 Then the rational number has the infinite exansion ( 2 (.4) =, 3, 4,... 2 3 More generally, if β,r N, then, if r+ > β, the rational number r has the exansion β ( ) (.5) r r+ β = β, r+2 β, r+3 β,.... r r+ r+2 These exansions follow by induction from the Engel algorithm above with A = A = r A n = n+r β, and a n = r+n β r+n. It turns out that rational numbers with infinite Engel exansions all have digits that ultimately follow the attern in (.5). Theorem. Let A Z \{0}. Then A is rational, A = α, if and only if either the Engel β exansion of A is finite, or there exists an n and an r n, such that ). (.6) a n+j = r+j+ γ r+j for j = 0,,2,... β,

where γ β. ENGEL EXPANSIONS 3 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,..., }. We now state our main results Theorem 2. The following assertions hold: (i) The valuations of the Engel-digits a n obey a law of large numbers; more recisely, for almost all x I v (a n ) lim = n n. (ii) The valuations of the Engel-digits a n obey a central limit theorem: [ lim P x I : v (a n )+ n < t = t e u2 /2 du. n n/( ) 2π (iii) For almost all x I, lim su n v (a n+ (x))+v (a n (x)) log n =, and lim inf ( v (a n+ (x))+v (a n (x))) =. n (iv) v (x n ) obeys a law of large number; more recisely, for almost all x I, ( n 2v x ) n 2( ), as n. (v) v (x n ) obeys central limit theorem ) v (x n lim P x I : n V(v (+ )) where V(v (+ )) = (n+)(n+2)(2n+3) 6 (n+)(n+2) 2 ( ) 2. < t = t 2π In articular we see from (i) that for almost all x I, a n /n e u2 /2 du., as n. Regarding (i), (ii), (iv), and (v) above we note the similar but weaker results shown in [5 holding for all x in I, v (a n ) n and x n (n+)(n+2) 2, n =,2,3,....

4 P. J. GRABNER AND A. KNOPFMACHER Furthermore, we consider the random variables a r+(x) a r(x) r, r =,2,3,.... These are indeendent and identically distributed with infinite exectation. However, the following result holds. Theorem 3. For any fixed ε > 0, i.e. lim P n n nlog n r= a r+(x) a r(x) [ x I : a r+ (x) nlog n a r= r (x) ( ) > ε = 0, ( ) in robability over I. Remark. Since a theorem in Galambos [3 (. 46), imlies that either lim su a r+ (x) n nlog n a r (x) = a.e. or lim inf n nlog n r= a r+ (x) a r (x) = 0 a.e., the conclusion of Theorem 3 does not carry over to validity with robability one. The aer is organized into sections, which slit the roofs of the theorems. Section 2 treats rationality criteria and the roof of Theorem. Section 3 gives some elementary robabilities, which will be used in the subsequent roofs, Section 4 gives the roof of Theorem 2 and Section 5 gives the roof of Theorem 3. 2. Rationality Criteria (Proof of Theorem ) Firstly, if A Z \{0} has an infinite Engel exansion, which satisfies (.6) then n ( A = + + ) + a k= a k a a n a n a n a n+ n ) = + ( r Q, a a k a a n β k= using (.5) and the uniqueness of the Engel exansion. Now suose A = α Q. By the algorithm for each n, for which A β n 0, A n+ = a n A n and it follows that A n Q for each n. We will make use of the elementary inequality (2.) 0 < a n < ( ) k = for all a n S. k=0 By (2.) we can write a n = b n v(an), where b n N and (2.2) 0 < b n < v(an)+.

ENGEL EXPANSIONS 5 Furthermore, each A n can be reresented in the form A n = αn β n v(an), where α n Z, β n N, (α n,β n ) =, and α n β n. An analogous reresentation holds also for A n+, rovided A n+ 0. Substituting these reresentations into (.2) leads to (2.3) α n+ v(a n+) β n+ = b nα n β n β n. Since (α n+ v(a n+),β n+ ) = it follows that β n+ β n. Thus by (2.) and using v (a n ) n we have (2.4) α n+ v(a n+) (b n α n +β n ) < v(a n+) c (a n)+ α n + v(a n+) β n < α n +β n n. By choosing N large enough, so that β N <, we have for all n N that α n+ α n. Suose that α n 0 for all n. Then for all n sufficiently large we have α n = α and β n = γ, where α,γ N and γ β. Substituting this into (2.3) yields which imlies that ± α v(a n+) β = ±b nα γ γ ±b n = γ α ± v(a n+). Since b n N and (α,γ) =, we must have α =. In the case that α n = + we get that b n = γ ± v(a n+). Since v (a n+ ) n+ and b n > 0 we conclude that b n = γ + v(a n+) γ + v(an) > v(an), which contradicts (2.2). In the case α n = it follows from (2.3) that α n+ = as well and hence b n = v(a n+) γ < v(an) only if v (a n+ ) = v (a n ) for n N, since γ v(a N) <. Consequently, in order for an infinite Engel exansion to exist it is necessary that α N+j = (j = 0,,...), v (a N+j ) = v (a N ) j, A N+j = v(a N ) γ, and a N+j = b N+j v(a N+j) = v(a N )+j+ γ v(a N )+j, which roves (.6) with n = N and r = v (a N ) n. Remark 2. If there exists m N such that A n < 0 then by (2.3) A n+j < 0 for every j. In articular, this imlies that every negative rational number A Z has an infinite -adic Engel exansion of tye (.6). Remark 3. For ositive rational numbers A Z, both terminating and infinite exansions can occur. However, in the secial case, when A N we see by induction that A n 0 for all n and it then follows from the roof of Theorem that every ositive integer A has a finite Engel exansion. 3. Basic Probabilities We begin by deriving some basic robabilistic results concerning the digits in -adic Engel exansions.

6 P. J. GRABNER AND A. KNOPFMACHER Lemma. The digits a n S form a Markov chain with initial robabilities (3.) P[v (a ) = j = ( ) j, and transition robabilities (3.2) P [ v (a n+ ) = k v (a n ) = j { ( ) j k for k > j = 0 otherwise. Proof. Firstly by the Engel algorithm A = x I. Then using the definition of Haar measure P [ v (A ) > j = P[v (a ) < j = j. Thus P[v (a ) = j = P[v (a ) < (j ) P[v (a ) < j = ( ) j. Next, A 2 is obtained from A by a system of linear congruences to successively higher owers of, arising from the relation A 2 = a A. From this it follows that A 2 is uniformly distributed in j I where j = v (a ). Inductively, if v (a n ) = j then A n+ is uniformly distributed in j I for all n >. Since the event v (a n+ ) < k under the conditionthatv (a n ) = j isjustacylindersetintheinfiniteroductsace{0,..., } N, which is described by fixing k j of the digits in the -adic exansion of A 2 equal to 0, we conclude that P [ v (a n+ ) < k v (a n ) = j = j k and (3.2) follows immediately. Remark 4. Since the robability in (3.2) deends only on the difference k j this imlies that the random variables v (a n ) v (a n+ ) are indeendent and identically distributed. Thus for n < n 2 < < n j and k i, i =,2,...j, [ P v (a nj +) = v (a nj ) k j,v (a nj +) = v (a nj ) k j,... v (a n +) = v (a n ) k = ( ) j (k + +k j ). (3.3) Corollary 4. Let n = n (x) denote the random variable v (a n ) v (a n+ ), with 0 = v (a ). Then [ P # { l n l = } ( )( n = k = k k ) k n. Thus the number of times that( degrees ) of consecutive digits increase by has a binomial distribution with mean value n and variance n. 2 In articular the lim inf result of art (ii) of Theorem 2 follows immediately. Corollary 5. The random variables n have mean value and variance E( n ) = and V( n ) = ( ) 2.

ENGEL EXPANSIONS 7 Proof. By Lemma Similarly E( n ) = lp[v (a n ) v (a n+ ) = l = ( ) l l =. l= E( 2 n ) = ( ) l= from which the formula for V( n ) is immediate. Lemma 2. The following equations hold: l= l 2 l = +2 ( ) 2 (i) ( ) t (3.4) P[v (a n ) = t = ( ) n t n and therefore (3.5) P[ n : v (a n ) = t =. (ii) P[v (a n+m ) = t ( ) t s v (a n ) = s = ( ) m s t. m Proof. First we rove statement (i). Since the sequence of degrees of the digits a,a 2,... is strictly increasing we have by Lemma, P[v (a n ) = t = j <j 2 < <j n <t P [ v (a n ) = t v (a n ) = j n P [ [ v (a n ) = j v n (a n 2 ) = j n P v (a 2 ) = j v 2 (a ) = j P[v (a ) = j = ( ) n jn t j n 2 j n... j j 2 j, j <j 2 < <j n <t = ( ) n t j <j 2 < <j n <t ( ) t = ( ) n t n. Thus we have ( ) t P[ n : v (a n ) = t = ( ) n t n n= t ( ) t = ( ) t ( ) l = l. l=0

8 P. J. GRABNER AND A. KNOPFMACHER For the roof of (ii) we find P [ v (a n+m ) = t v (a n ) = s = s<j <j 2 < <j m <t P [ v (a n+m ) = t v (a n+m ) = j m P [ [ v (a n+2 ) = j v 2 (a n+ ) = j P v (a n+ ) = j v (a n ) = s = ( ) m s t s<j <j 2 < <j m <t = ( ) m s t ( t s m ). Remark 5. From the roof of (i) we can also deduce the joint robability distribution P[v (a ) = j,...,v (a n ) = j n = ( ) n jn, rovided that the growth condition v (a i ) i holds for each i =,2,...,n. Otherwise the joint robability distribution has value 0. Lemma 3. Let X n be a sequence of indeendent, identically distributed random variables with EX n = µ and VX n = σ 2. Then (3.6) lim n 2 n(n+) (n+ k)x k = µ almost surely. Proof. Under these hyotheses the law of large numbers lim X k = µ almost surely n n k= k= holds. Since (3.6) is just the second order Césaro mean of the random variables X k. Since the first order Césaro mean exists almost surely and equals µ, so does the second order mean. 4. Proof of Theorem 2 Since we can write v (a n ) as the sum of indeendent random variables n n v (a n ) = (v (a i+ ) v (a i ))+v (a ) = i, i= it follows from Corollary 2 that v (a n ) has mean and variance E(v (a n )) = n. and resectively. V(v (a n )) = n ( ) 2, i=0

ENGEL EXPANSIONS 9 Hence by the law of large numbers and the central limit theorem (see e.g. Feller [2,. 244, 253) arts (i) and (ii) of Theorem 2 follow. For the roof of (iii) we note that the events v (a n ) v (a n+ ) > k(n) are indeendent with robabilities P [ n > k(n) = k(n). The Borel-Cantelli lemmas then yield P [ n > k(n) for infinitely many n { 0, if n= = k(n) converges if n= k(n) diverges. By choosing k(n) = clog n we see that with robability the events (v(an) v(a n+)) > c log n occur infinitely often if c and only finitely often if c >. The limsu result then follows. The corresonding lim inf result was already shown in Section 2. (iv), (v) We first comute the mean and variance of v (x n ). In [5 it is shown that x n = v(qn+). Now n+ E(v (+ )) = r= E(v (a n )) = To comute the variance we make use of the fact that (4.) (4.2) n+ n+ r v (+ ) = v (a r ) = r= = r= l (n+ l). l=0 (n+)(n+2) 2 We now remark that the last sum has the same distribution as the sum (l +) l. Thus we have for the variance V(v (+ )) = (l+) 2 V l = (n+)(n+2)(2n+3) 6 l=0 l=0 l=0 l. ( ) 2. Assertion (iv) now follows form Lemma 3. For the roof of (v) we check that the random variables (l + ) l satisfy Lindeberg s condition (cf. [2,.256): since s 2 n = V(v (+ )) is of order of magnitude n 3, we have to comute the integrals 2 df k (y) = (k +) 2 x 2 df(x) (k +) 2 y tn 3/2 y x tn3/2 k+ x 2 df(x), x t 2 n

0 P. J. GRABNER AND A. KNOPFMACHER where F k is the distribution function of (k + )( k ) and F = F 0. Thus the last integral is equal to the sum k + t 2 n ( k ) 2 k = O( n t ) 2 n for n sufficiently large, and we have ( y 2 df s 2 k (y) = O n k=0 y ts n for any t > 0. Thus n (n+)(n+2) 2 ) t 2 n 0 v (+ ) V(v (+ )) has asymtotically normal distribution and the roof is comleted. 5. Proof of Theorem 3 We first notice that by Lemma the random variables a r+(x) a r(x) r are indeendent and identically distributed with infinite exectation. We write s = log y iff y = s and use the truncation method of Feller [2, Chater 0, 2, alied to the random variables U r,v r (r n) defined by Then (5.) (5.2) (5.3) U r (x) = ar+ / ar (x),v r (x) = 0 if ar+ / ar (x) nlog n, U r (x) = 0,V r (x) = ar+ / ar (x) if ar+ / ar (x) > nlog n [ P x I : a r+ (x) nlog n a r= r (x) ( ) > ε U P[ + +U n ( )nlog n > εnlog n + +P [ V + +V n 0, and using Lemma, [ a 2 (x) (5.4) P V + +V n 0 np[ a (x) > nlog n (5.5) = n Now note that k k >nlog n ( ) k log n = o(). E(U + +U n ) = ne(u ), V(U + +U n ) = nv(u ),

where (5.6) (5.7) and E(U ) = a 2 (x) a (x) nlog n ENGEL EXPANSIONS k P[ = k = = ( )log ( [nlog n ), V(U ) < E(U 2 ) = k nlog n k nlog n k ( ) k ( ) k < qnlog n. Chebyshev s inequality then yields U P[ + +U n ne(u ) (5.8) > εne(u ) (5.9) nv(u ) ( εne(u ) ) 2 < n 2 log n (ε( )nlog ( [nlog n )) 2 = o(). Since E(U ) ( )log n as n, Theorem 2 follows. References [ P. Erdős, A. Rényi, and P. Szüsz. On Engel s and Sylvester s series. Ann. Univ. Sci. Budaest. Eötvös. Sect. Math., :7 32, 958. [2 W. Feller. An Introduction to Probability Theory and its Alications, volume 2. J. Wiley, 966. [3 J. Galambos. Reresentations of Real Numbers by Infinite Series, volume 502 of Lecture Notes in Mathematics. Sringer-Verlag, Berlin, 976. [4 P. J. Grabner and A. Knofmacher. Metric roerties of Engel series exansions of Laurent series. Math. Slovaca, 48:233 243, 998. [5 A. Knofmacher and J. Knofmacher. Series exansions in -adic and other non-archimedean fields. J. Number Theory, 32:297 306, 989. [6 A. Knofmacher and J. Knofmacher. Infinite series exansions for -adic numbers. J. Number Theory, 4:3 45, 992. [7 A. Knofmacher and J. Knofmacher. Metric roerties of some secial -adic series exansions. Acta Arith., 76: 9, 996. [8 N. Koblitz. -Adic Numbers, -Adic Analysis, and Zeta-Functions. Sringer-Verlag, New York, 984. [9 V. Laohakosol. A characterization of rational numbers by -adic Ruban continued fractions. J. Austral. Math. Soc. Ser. A, 39:300 305, 985. [0 O. Perron. Irrationalzahlen. Walter de Gruyter & Co., Berlin, 960. [ A. A. Ruban. Certain metric roerties of the -adic numbers. Sibirsk. Mat. Ž., :222 227, 970. [2 W. H. Schikhof. Ultrametric calculus. Cambridge University Press, Cambridge, 984. An introduction to -adic analysis. (P. G.) Institut für Mathematik A, Technische Universität Graz, Steyrergasse 30, 800 Graz, Austria E-mail address: eter.grabner@tugraz.at (A. K.) The John Knofmacher Centre for Alicable Analysis and Number Theory, University of the Witwatersrand, Private Bag 3, WITS 2050, South Africa E-mail address: arnoldk@cam.wits.ac.za URL: htt://www.wits.ac.za/science/number theory/arnold.htm