On Error Sum Functions Formed by Convergents of Real Numbers

Transcription

1 Journl of Integer Sequences, Vol. 4 (), Article.8.6 On Error Sum Functions Formed by Convergents of Rel Numbers Crsten Elsner nd Mrtin Stein Fchhochschule für die Wirtschft Hnnover Freundllee Hnnover Germny Crsten.Elsner@fhdw.de Abstrct Let p m /q m denote the m-th convergent (m ) from the continued frction expnsion of some rel number α. We continue our work on error sum functions defined by E(α) := m q mα p m nd E (α) := m (q mα p m ) by proving new density result for the vlues of E nd E. Moreover, we study the function E with respect to continuity nd compute the integrl E(α)dα. We lso consider generlized error sum functions for the pproximtion with lgebric numbers of bounded degrees in the sense of Mhler. Introduction nd sttement of the min results Recently the first uthor [] introduced two error sums: Let α = [ ;,,...] be the continued frction expnsion of rel number α, which my be finite in the cse of rtionl number α. Let p m q m = [ ;,..., m ] (m ) denote the convergents of α. The error sum functions E(α) nd E (α) re defined by E(α) = m E (α) = m (αq m p m ). αq m p m = m ( ) m (αq m p m ),

2 Both functions do not depend on the integer prt of α. So we my restrict their domins on the intervl [, ). The first uthor [] proved tht E(α) ρ = + 5 nd E (α) (α R). The series m q mα p m [,ρ] mesures the pproximtion properties of α on verge. The smller this series is, the better rtionl pproximtions α hs. Nevertheless, α cn be Liouville number nd m q mα p m tkes vlue close to ρ. So, it my be interesting to question on the verge vlue of E nd E, respectively. We compute the verge vlue of E, see Theorem 5. The error sum functions E nd E hve vrious interesting properties. In [], pplictions re discussed for certin trnscendentl numbers nd for qudrtic irrtionl numbers. For instnce, we hve E(exp()) = m q m e p m = e E (exp()) = m (q m e p m ) = E( 7) = m q m 7 pm = exp( t )dt e = , exp(t )dt e + 3 = , = , E ( 7) = (q m 7 pm ) = m = It is cler tht for ny rtionl number α the series for E(α) nd E (α) become finite sums nd therefore belong to Q. In the cse of qudrtic irrtionl numbers α we hve E(α) Q(α) nd E (α) Q(α) ([, Theorem 3]). But for qudrtic irrtionls E(α) Q(α) \ Q does not hold in generl. For exmple, E((3 5)/) = (see [3, Lemm 8]). On the other hnd E(α) Q(α) is not true for ll rel numbers α. For α = e = exp() we hve E(e) Q(e), since e nd exp( t )dt re lgebriclly independent over Q. This follows from remrk on pge 93 in [8]. Similrly, one cn show tht E (e) Q(e). The uthors [3] studied the vlue distribution of the error sum functions in more detil. They constructed two lgorithms which prove tht the set of vlues of E is dense in the intervl I E = [,ρ], nd tht the set of vlues of E is dense in the intervl I E = [, ] (see [3, Theorems, ]). But, given ny uniformly modulo one distributed sequence (α ν ) ν of rel numbers, the sequences (E(α ν )) ν nd (E (α ν )) ν re not uniformly distributed in I E nd I E, respectively (see [3, Theorems 3, 4]). In this pper we show tht ny dense subset of (, ) is mpped by E(α) nd E (α) into set which is dense in I E nd I E, respectively. Then, we continue to study the nlytic properties of the error sum functions. The function E hs lredy been investigted by Ridley nd Petrusk [7]. Among other things they showed tht E (α) is continuous t every irrtionl point α, nd discontinuous when α is rtionl. Moreover, they computed the integrl E (α)dα by pplying the functionl eqution E (α) + E ( α) = mx{α, α} except t α = nd α =.

3 Inspired by the work of Ridley nd Petrusk, we prove similr results for the error sum function E. We compute the integrl E(α)dα by using multiple sum, which expresses the integrl in terms of denomintors of convergents. Unfortuntely, the functionl eqution { α, if < α < /; E(α) E( α) = α, if / < α < ; cnnot be used to evlute the integrl E(α)dα. The min results of this pper re given by the following theorems. Theorem. Let (α n ) n be sequence of rel numbers forming dense set {α n : n N} in (, ). Then the set {E(α n ) : n N} is dense in (,ρ), nd the set {E (α n ) : n N} is dense in (, ). Theorem. The function E(α) is discontinuous t every rtionl point α, nd it is continuous t every irrtionl point α. Exmple 3. Let n, k be integers with n, k 3. For x = /n we hve ( E n + ) = n k n + 3 (k ), n k n ( E n ) = n k n n + (k ), k n k n ( E n + ) = n k n (k ), k n ( k E n ) = n k n 3 (k ). n k n These expressions re obtined by using the identities n + n k = [ ;n,,n k,n], n n k = [ ;n,n k,,n ]. Let m, nd let,..., m be positive integers. Set p m q m = [ ;,..., m ], where p m nd q m with q m > re coprime integers. Theorem 4. We hve E(α)dα = + m= = m= q m (q m + q m ), nd E (α)dα = + m= = m= ( ) m q m (q m + q m ). 3

4 With the first identity from the preceding theorem, we compute the men vlue of the function E. Theorem 5. We hve E(α)dα = ζ() log ζ(3) = , where ζ(s) denotes the Riemnn zet function. Remrk 6. Ridley nd Petrusk [7] proved tht E (α)dα = 3 8. We point out tht by Theorem5 nd Remrk6 the men vlues of E nd E re less thn hlf of the mximum vlue of E nd E, respectively. In Section 5 we generlize the error sum function E to the pproximtion with lgebric numbers of bounded degree. Here, the Mhler function w n (H,α) will be involved. Proof of Theorem We will only prove the sttement concerning the vlues of the function E, since there re no dditionl rguments for the function E. It is shown in the proof of Theorem in [3] tht the set {E(α) : α Q (, )} is dense in (,ρ). Hence, for ny rel number η (,ρ) nd for ny δ > there is rtionl number r (, ) stisfying η E(r) < δ 3. () By r = [;,,..., t ] = p t q t we denote the continued frction expnsion of r. Without loss of generlity we my ssume tht t stisfies + ( ) t < δ 3. () This cn be seen by the following rgument: For ny number r = [;,..., t ] stisfying η E(r ) < δ/3 nd t < t we construct number r = [;,,..., t ] with t + = = t = b, such tht t stisfies () nd b is sufficiently lrge (see [3, Lemm ]). Nmely, for r k 4

5 defined by r k := [;,..., t,b,...,b] we hve }{{} k E(r) E(r ) = E(r t t ) E(r ) = t t E(r k+ ) E(r k ) < t b t t k= k= E(r k+ ) E(r k ) < (b ). t t k= b = t t b Since the set {α n : n N} is dense in (, ) by the ssumption in the theorem, there is positive integer m stisfying nd α m = [;,,..., t, t+,...] r α m < δ 3(t + )q t. (3) Let p ν /q ν be the convergents of α m. Then, by pplying the inequlities (), (3) nd () we hve η E(α m ) = η E(r) + E(r) E(α m ) η E(r) + E(r) E(α m ) < δ 3 + t q ν r p ν q ν α m p ν ν ν= δ t 3 + r α m q t + q ν α m p ν ν= δ 3 + t ν= δ 3 + ( ) ν ν t = δ ( ) t < δ, ν t+ δ 3(t + ) + q ν t+ ν which completes the proof of Theorem. 3 Proof of Theorem Since the function E is periodic of period one, it suffices to prove Theorem for α [, ). We will prove the sttement on continuity first. Let η [, ) be rel irrtionl number, 5

6 sy η = [;,,...], nd let (ξ n ) n be sequence of rel numbers converging to η. By I m = I m (,..., m ) we denote the intervl defined uniquely by [;b,b,...] I m (b = b m = m ). (4) The boundry points of I m re rtionl numbers, nd therefore the irrtionl number η lies in the interior of I m for ny m. With lim n ξ n = η we conclude on ξ n I m (n n ) for some positive integer n = n (m). Hence, by (4), we hve Let p ν /q ν for ν be the convergents of η nd let p (n) ν from (5), it follows tht p ν ξ n = [;,..., m,...]. (5) = p(n) ν q ν q ν (n) /q ν (n) ( ν m). For fixed positive integer m nd ny n n we estimte E(η) E(ξ n ) = q ν η p ν q ν (n) ξ n p (n) ν ν ν m ( ) ν q ν (η ξ n ) + q ν η p ν + = ν= ν m+ m ( ) ν q ν (η ξ n ) + ν= ν m+ m ( ) ν q ν (η ξ n ) + ν= ν= ν m+ + q ν be the convergents of ξ n. Then, ν m+ ν m+ q (n) ν + (ν )/ m ( ) ν q ν (η ξ n ) + ( ) m. ν m+ q ν (n) ξ n p (n) ν (ν )/ Since m cn be chosen rbitrry lrge nd ξ n tends to η for incresing n, we conclude on lim E(ξ n) = E(η). n This proves tht the function E(α) is continuous t every irrtionl point α. To prove the sttement on discontinuity we shll t first discuss the cse when η is rtionl number in (, ). Let η = [;,,..., m ] 6

7 for some integers m nd m >. Moreover, let (ξ () n ) n nd (ξ () n ) n be two sequences of rtionls defined by Obviously we hve ξ () n = [;,..., m,n] nd ξ () n = [;,..., m,,n] (n ). lim n ξ() n = η = lim ξ n (). (6) n Let p () ν /q ν () for ν =,...,m+ be the convergents of ξ n (). By p () ν /q ν () we denote the convergents of ξ n (). Then we hve for ν =,...,m+ p () ν q () ν = p() ν q ν () ( ν m ). Therefore we my set p ν := p () ν = p () ν nd q ν := q ν () = q ν () for ν =,...,m. We compute m E(ξ n () ) E(ξ n () ) ( ) ν (ξ n () ξ n () )q ν ν= = ( ) m (( m )q m + q m )ξ n () ( ) m (( m )p m + p m ) + ( ) m+ ( m q m + q m )ξ n () ( ) m+ ( m p m + p m ) ( ) m ( m q m + q m )ξ n () + ( ) m ( m p m + p m ) = ( ) m (ξ n () ξ n () )( m q m + q m ) + ( ) m (p m q m ξ n () ) + ( ) m+ ( m q m + q m )ξ n () ( ) m+ ( m p m + p m ). For n, by (6) nd with η = p () m /q m () lim n we obtin the limit ( E(ξ () n ) E(ξ n () ) ) = ( ) [ m (p m q m η) + (p () m q m () η) ] = ( ) mp() m q m () p () m q () m q m () =. q m () In prticulr, by /q m (), this proves tht the function E is discontinuous t η. It remins to prove tht E is discontinuous t η =. Let ξ n () := [;n] nd ξ n () := [ ;,n]. Then both sequences (ξ n () ) n nd (ξ n () ) n tend to for incresing n, but E(ξ () n ) = n (n ), whers E(ξ () n ) = holds for every positive integer n. Hence, Theorem is proven. 4 Proofs of Theorem 4 nd Theorem 5 Proof of Theorem 4. Let m nd,..., m be positive integers. Set ξ = [;,..., m, m ], ξ = [;,..., m, m + ]. 7

8 Then we hve ξ < ξ for even m nd ξ < ξ otherwise. We define I m := (ξ,ξ ) for even m nd I m := (ξ,ξ ) for odd m, which depend on,..., m. The intervls I m re disjoint for different m-tuples (,..., m ). For ny fixed m the union of ll closed intervls I m gives the intervl [, ]. With this decomposition of [, ] we obtin E(α)dα = ( ) m (q m α p m )dα nd = E (α)dα = m= ( ) m (q m α p m )dα m= = + ( ) m m= = + m= = = m= (q m α p m )dα m= = + m= = = + ( ) m m= m= = m= ξ I m (q m α p m )dα ξ (q m α p m )dα (7) (q m α p m )dα I m ξ (q m α p m )dα ξ (8) Every point α I m stisfies α = [;,..., m, m,...], hence the convergents p ν /q ν for ν m depend on I m, but not on α I m. Therefore, we derive Using ξ m= (q m α p m )dα = (ξ ξ ) (ξ + ξ )q m p m ξ ξ = p m nd ξ = ( m + )p m + p m q m ( m + )q m + q m we compute the expressions ( ) m ξ ξ = (q m + q m )q m nd ξ + ξ = p m q m + q m p m + p m q m, (q m + q m )q m which give ξ (q m α p m )dα = ξ q m (q m + q m ). Substituting this integrl into (7) nd (8), we finlly get the formuls stted in the theorem.. 8

9 Proof of Theorem 5. First we show tht + m= = m= q m (q m + q m ) = = b= gcd(,b)= ( + b). (9) For the denomintors of two subsequent convergents of the continued frction expnsion of α = [;,..., m,...] it is well-known tht gcd(q m,q m ) =. For fixed q m = we count the solutions of q m = b with gcd(,b) = nd b in the multiple sum on the left-hnd side of (9). It is necessry to distinguish the cses m nd m =. Cse : m. First let =. Then, For we hve q m q m = [; m,...,, ] = [; m,..., + ]. q m q m = [; m,...,, ] = [; m,...,,, ]. Cse : m =. For = we hve unique representtion of the frction q m q m = q q = = = [; ], since the integer prt = must not be chnged. For there re gin two representtions: q m = q = = [; ] = [;, ]. q m q Therefore it is cler tht for fixed q m = every b with gcd(,b) = nd b occurs exctly two times in the multiple sum on the left-hnd side of (9), except for m = nd =. For this exceptionl cse we seprte the term q (q + q ) = 8 9

10 from the multiple sum. Then we obtin + m= = m= q m (q m + q m ) = + = + m= = = = + = = = b= gcd(,b)= b= gcd(,b)= m= b= gcd(,b)= q m (q m + q m ) + ( + b) + 8 ( + b) + 8 ( + b), = q (q + ) + 8 which proves the identity in (9). Next we tret the double sum on the right-hnd side of (9). Let µ denote the Möbius function. Then we derive = b= gcd(,b)= ( + b) = = = d= b= = d b= d b d> d gcd(,b) µ(d) ( + b) = µ(d) ( + b) = = b= n /d d= n= m= d> d d b µ(d) ( + b) µ(d) nd(nd + md) = d= = ζ(3) = ζ(3) = ζ(3) = µ(d) d 3 = c= c= n n= m= n(n + m) = ζ(3) c = ζ(3) c c ζ(, ) ζ(3) c= c ( ) + = c ( ) + = + 3 4, c= c + ζ(3) = b= c = c/ + ( ) c+ c= c 3 ( + b)

11 where ζ(, ) = c= c c = ( ) = c>> ( ) c is specil cse of the multivrite zet function (see [, Section.6]), stisfying Collecting together we obtin from (9) tht ζ(, ) = ζ(3) 3 ζ() log. E(α)dα = ζ() log = 5 3ζ() log +, ζ(3) 8 ζ(3) which completes the proof of the theorem. Remrk 7. Let n be positive integer. We consider modified error sum function given by αq m p m n ( < α < ). m By similr methods s used to deduce Theorems 4 nd 5 we obtin the following identities: m αq m p m n dα = n + + n + ( = n + m= = m= ζ(n +, ) n+ ζ(n + ) with the multivrite zet function ζ(n +, ) defined by ζ(n +, ) = m >m > ( ) m m m n+. q m (q m + q m ) n+ ) This yields n symptotic expnsion, nmely m αq m p m n dα = n + + O ( ) (n + ) n (n ). 5 Generliztion of the error sum function E In this section we show tht the error sum function E is the specil cse of more generl concept involving the theory of pproximtion with lgebric numbers of bounded degree. We need some nottions to recll the definition of the Mhler functions w n (H,α) nd w n (α). For more detils on this function we refer to [5]. For ny polynomil P(x) Z[x] we denote by H(P) the height of the polynomil P, which is given by the mximum vlue of the modulus of the coefficients. Let n,h be positive integers

12 nd α C with α / nd deg α > n. For α being trnscendentl we define deg α =. Set w n (H,α) := min P(α), P Z[x] \ {} deg P n H(P) H w n (α) := lim sup H log w n (H,α) log H w n (α) is the lrgest positive rel number such tht for every ε > there re infinitely mny polynomils P from Z[x] of degree t most n stisfying P(α) < ( H(P) ) w n(α)+ε. So the function w n (H,α) is needed to define the importnt Mhler function w n (α). From the definition of w n (H,α) it follows immeditely tht w (H,α) w (H,α) w n (H,α) holds for ll integers n =,,... Given α nd some positive integer n with deg α > n, there is unique sequence (H m ) m of positive integers stisfying the following conditions: (i) = H < H < < H m <... (ii) w n (H,α) > w n (H,α) > > w n (H m,α) >... (iii) w n (H m,α) = w n (H m+,α) (m =,,...) We define the generlized error sum function E n (α) := w n (H m,α). m= Note tht E n (α) = E n ( α) holds, since the sme is obviously true for the Mhler function: w n (H,α) = w n (H, α). For n = nd α ( /, /) \ Q we hve p /q { /, /} nd p /q = /, where = holds if nd only if / < α <. This implies tht { qm α p w (H m,α) = m, if < α < /;, (m =,,...). q m+ α p m+, if / < α < ; Therefore, { E(α), if < α < /;, E (α) = E(α) α, if / < α < ;, where α + equls q α p in the second cse. Let E n := sup { E n (α) : α ( /, /) deg α > n } (n =,,...). Then it is cler tht for n =,,... { } E n = sup w n (H m,α) : α ( /, /) deg α > n m= { sup m= } w (H m,α) : α ( /, /) deg α > n E = sup { E (α) : α ( /, /) deg α > } ρ..

13 This bound cn be improved by pplying two inequlities bsed on Siegel s Lemm. Let α C with α < /. For rel α nd ny positive integers n,h we hve For α / R nd ny positive integers n,h we hve w n (H,α) < (n + )H n. () w n (H,α) < (n + )H (n )/. () These inequlities cn be found on pge 69 in [5], where the constnts C nd C re given by [5, Hilfsstz 7, Hilfsstz 8]. In wht follows we distinguish whether α is rel or not. Cse : α R. By using w n (,α) mx x n = / x / n we obtin with () nd the Riemnn zet function for n E n (α) = w n (H m,α) = w n (,α) + m= m= w n (H m,α) m= + w n n (m +,α) + n + n (m + ) n m= = n + (n + )( ζ(n) ) (for n ). Cse : α / R. Here we consider the polynomil z n for z /. Then, w n (,α) mx z / zn = n. With () we repet the rguments from Cse for n 4: (n + ) E n (α) w n (,α) + (m + ) (n )/ m= + ( ( ) ) n (n + ) ζ n (for n ). Note tht the inequlity holds for n 3, wheres n + (n + )( ζ(n) ) < ρ + ( (n + ) ζ n ( ) n ) < ρ is true for n 5. 3

14 6 Acknowledgments The uthors re much obliged to Mr. H. A. ShhAli nd Professor A. Ustinov. Mr. ShhAli hs drwn our ttention to the pper [7] of J. N. Ridley nd G. Petrusk. Professor A. Ustinov, who joined the second uthor t the Journées Arithmétiques in Vilnius, hs given useful hints to compute the vlue of the integrl in Theorem 5. References [] D. H. Biley, J. M. Borwein, N. J. Clkin, R. Girgensohn, D. R. Luke, nd V. H. Moll, Experimentl Mthemtics in Action, A. K. Peters, 7. [] C. Elsner, Series of error terms for rtionl pproximtions of irrtionl numbers, J. Integer Sequences 4 (), Article..4. [3] C. Elsner nd M. Stein, On the vlue distribution of error sums for pproximtions with rtionl numbers, submitted. [4] A. Khintchine, Kettenbrüche, Teubner, 956. [5] Th. Schneider, Einführung in die trnszendenten Zhlen, Springer-Verlg, 957. [6] O. Perron, Die Lehre von den Kettenbrüchen, Chelse Publishing Compny, 99. [7] J. N. Ridley nd G. Petrusk, The error-sum function of continued frctions, Indg. Mthem., N.S., (),, [8] A. B. Shidlovskii, Trnscendentl Numbers, de Gruyter, 989. Mthemtics Subject Clssifiction: Primry J4; Secondry J7, B5, B39. Keywords: continued frctions, convergents, pproximtion of rel numbers, error terms, density. (Concerned with sequences A45, A7676, A7677, A48, nd A49.) Received My 3 ; revised versions received July ; August 7. Published in Journl of Integer Sequences, September 5. Return to Journl of Integer Sequences home pge. 4