IRRATIONALITY MEASURES IRRATIONALITY BASES AND A THEOREM OF JARNÍK JONATHAN SONDOW ABSTRACT. We recall that the irratioality expoet µα ( ) of a irratioal umber α is defied usig the irratioality measure µ. Usig β istead ad motivated by a coditioal result o Euler's costat we defie the irratioality base βα ( ). We give formulas for µα ( ) ad βα ( ) i terms of the cotiued fractio expasio of α. The formulas yield bouds o µα ( ) ad βα ( ) ivolvig the Fiboacci umbers as well as a growth coditio o the partial uotiets of α implyig µα ( )= ad a weaker coditio implyig βα ( )=. A theorem of Jarík o Diophatie approximatio leads to umbers with prescribed irratioality measure. By a differet method we explicitly costruct series with prescribed irratioality base. May examples are give.. INTRODUCTION Recall that the irratioality expoet µα ( ) of a irratioal umber α is defied i terms of the irratioality measure µ. Usig β istead we itroduce a weaker measure of irratioality the irratioality base βα ( ) as follows. If there exists a real umber β with the property that for ay ε > 0 there is a positive iteger ( ε ) such that p α > ( β + ε) for all itegers p with ( ε ) the the least such β is called the irratioality base βα ( ) of α. The motivatio for usig the irratioality measure β rather tha some other fuctio of is the followig. I [6] we gave criteria for irratioality of Euler's costat γ ivolvig the Beukers-type double itegral where I : [ 0 ] ( x( x) y( y)) ( xy) log xy dx dy = γ + L A dl Zl( + ) + Zl( + ) + + Zl( )
ad d A Z with d deotig the least commo multiple of the umbers... I particular we showed that γ is irratioal if the fractioal part of d L exceeds ifiitely ofte. I a paper i preparatio we prove that a stroger coditio implies a measure of irratioality for γ. Namely if (C) lim l d L = 0 where t deotes the distace from t to the earest iteger the the estimates as o I = 4 ( + ( )) o = + ( ( )) d = e ( + o( )) imply that for ay ε > 0 there exists ( ε ) > 0 such that p γ > ( e + ε) for all itegers p with ( ε ); i other words the irratioality base of γ satisfies the ieuality βγ ( ) e = 5. 436.... We also show that a coditio similar to (C) but based o a itegral for l( π ) [7] implies that lπ is irratioal ad β(l π). Moreover we preset umerical evidece for both coditios. These results suggest that it would be iterestig to study the ew irratioality measure β by itself. I Sectio we defie irratioality measures expoets ad bases. I Sectio 3 we prove formulas for µα ( ) ad βα ( ) i terms of the cotiued fractio expasio of α ad compute several examples. The formulas yield bouds o µα ( ) ad βα ( ) ivolvig the Fiboacci umbers as well as a growth coditio o the partial uotiets of α implyig µα ( )= ad a weaker coditio implyig βα ( )=. I Sectio 4 we observe that the existece of umbers with prescribed irratioality measure follows from V. Jarík's 93 costructio of cotiued fractios with certai Diophatie approximatio properties [3]. I the fial sectio we use a differet method to costruct explicitly series with prescribed irratioality base. The two methods yield umbers with the same irratioality base but differet approximatio properties.. IRRATIONALITY MEASURES EXPONENTS AND BASES After recallig the defiitios of irratioality measure ad irratioality expoet we itroduce the otio of irratioality base. Throughout this sectio ad the ext α deotes a fixed but arbitrary irratioal umber. Defiitio. A irratioality measure is a fuctio f( x λ ) defied for x ad λ > 0 which takes values i the positive reals ad is strictly decreasig i both x ad λ. If there exists λ > 0 with the property that for ay ε > 0 there exists a positive iteger ( ε ) such that
3 p α > f( λ + ε) for all itegers p with ( ε ) the we deote by λα ( ) the least such λ ad we say that α has irratioality measure f( x λα ( )). Otherwise if o such λ exists we write λα ( )=. λ Defiitio. Let f( x λ) = x. If α has irratioality measure f( x µ ) = x for some µ = µ ( α) the µα ( ) [ ) is called the irratioality expoet of α. If there is o such µ the we write µα ( )= ad we call α a Liouville umber. Remark. Our defiitio of irratioality expoet is euivalet to the oe i [ p. 06] but our defiitio of irratioality measure differs from that i [ p. 7]. x Defiitio 3. Let f( x λ) = λ. If α has irratioality measure f( x β) = β for some β = β( α) (so that β is the smallest umber with the property that for ay ε > 0 there exists ( ε ) > 0 such that p () α > ( β + ε) for all itegers p with ( ε )) the we call βα ( ) [ ) the irratioality base of α. Otherwise if o such β exists we write βα ( )= ad we say that α is a super Liouville umber. Example. The sum of the followig series is a super Liouville umber: S = + + + 4. b Proof. If we write S = b where b = ad b = ( ) for > the the th partial sum of the series euals a b for some iteger a ad we have 0 < a S b < b = < < + b b. + ( ) ( ) b By the last boud S is irratioal. By the ext-to-last boud o umber β ca satisfy the reuiremets of Defiitio 3 with α = S. So β( S ) =. Writig ieuality () as p α > log( + β ε ) / log µ x
4 i order to compare it with the irratioality measure µ i Defiitio we see that a upper boud o the irratioality base of a umber is a much weaker coditio tha a upper boud o its irratioality expoet. I particular we have the followig propositio which justifies the termiology i the last part of Defiitio 3. Propositio. A irratioal umber which is ot a Liouville umber has irratioality base oe; euivaletly if β( α)> the µα ( )=. I particular a super Liouville umber is also a Liouville umber. Proof. If µ = µ ( α) is fiite the for ay ε > 0 we have α p > µ + ε > ( + ε) for all itegers p with > 0 sufficietly large ad so β( α)=. I particular βα ( )= implies µα ( )=. Example. The trascedetal umbers l e π ζ( ) ad the irratioal umber ζ( 3 ) all have irratioality base oe. Proof. Upper bouds o their irratioality expoets are kow [ Chapter 4 5]. Corollary. A algebraic irratioal umber has irratioality base oe. Proof. This is a immediate from Liouville's theorem. 3. FORMULAS FOR THE IRRATIONALITY EXPONENT AND BASE We prove formulas for the irratioality expoet ad base of a irratioal umber α i terms of its cotiued fractio expasio. The we give two corollaries ad several examples. The first corollary gives bouds o µα ( ) ad βα ( ) ivolvig the Fiboacci umbers. The secod gives two growth coditios o the partial uotiets of α : oe implies that µα ( )= ad the other weaker oe that βα ( )=. I this sectio ad the ext we will use the followig two lemmas. Lemma. (Legedre [5]) For itegers p with > 0 the ieuality p α < implies that p is a coverget of the cotiued fractio expasio of α. Proof. See [4 p. ].
5 Lemma. For ay real umbers C > 0 λ > there exists a positive iteger 0 = 0( C λ ) such that if p are itegers satisfyig the ieualities α p C λ 0 the p is a coverget of α. Proof. By Lemma it suffices to take 0 so large that C λ 0 < ( 0 ). Deote a simple cotiued fractio by [ b0; b b...] = b0 + b + b. + Theorem. The irratioality expoet ad irratioality base of a irratioal umber α with cotiued fractio expasio α = [ b0; b b...] ad covergets p are give by () µα ( ) limsup l + = + = +limsup l b l l + (3) l ( ) limsup l + βα = = limsup l b +. Remark. The formulas i () for the irratioality expoet are probably kow but we have ot foud a referece. Proof. If we defie λ by the euatio (4) α p = λ the the secod of the ieualities [4 p. 8] (5) p < α < + + implies that λ >. Usig (5) ad Lemma we deduce that
6 limsup λ = µ ( α ). Takig logarithms i (4) ad (5) leads to the first formula i () ad the secod follows usig the relatios [4 p. ] (6) + = b+ + = b+ ( + o( )). To prove (3) we defie λ by replacig the right-had side of (4) by λ. The (5) ad Lemma imply that limsup λ β( α ) ad (3) follows as did (). = Example 3. The golde mea φ = ( + 5)/ has irratioality expoet ad irratioality base. Proof. Here = F is asymptotic to φ ad so µφ ( )= + = ad l βφ ( ) = 0. Corollary. If α = [ b0; b b...] ad F is the -th Fiboacci umber the l b+ l b+ (7) limsup µα ( ) limsup l( F b b b ) l( F + b b b ) l b (8) limsup F b b + b l b+ l βα ( ) limsup. F + b b b Proof. Usig = b the recursio i (6) ad iductio (or the fact that is the sum of F products of the b i ) we obtai the bouds F + b b b F b b b ad (7) (8) follow usig () (3). Example 4. The umbers L 3 0!!! = [ ;...] L = [ 0 ; 4...] L 3 = [ 0 ;...] L 4 = [ 0 ; 3...] all have irratioality expoet ifiity ad irratioality base oe. I particular they are Liouville umbers but ot super Liouville umbers. O the other had the umbers
7 are super Liouville umbers. 3 S = [ 0 ; 3 3 4...] S = [ 0 ; 4 4...] Proof. We weake the lower boud i (7) ad the upper boud i (8) to The l b+ µα ( ) limsup l ( ) limsup l b+ βα l( bb b ) bb b ( + )! µ( ) limsup lim ( L )! +! +! + +! + = 3!. l ( ) limsup ( + )!l lim ( + β L )!! +! + +!! = 0. For L we have b = ( ) b ad so ( + ) b µ( L) limsup lim + + b + 3b + + b + b b = l ( ) limsup ( + β L )l + lim = 0 bb b. The estimates for L 3 ad L 4 are similar. For S use the tower otatio T by T k (a) T( k)= k T+ ( k ) ( ) = k to write b = T ( ). Sice ( k) defied by (8) we have T () T ( ) T ( ) < + 3 T ( ) T( + )l( + ) l β( S) limsup T () T ( ) T ( ) lim ( + ) T( + ) + T ( ) lim. = = + T The estimate for S where b = T ( ) is easier.
8 Remark. Ay oe of the umbers L L L3 L4 shows that the coverse of Propositio is false. Corollary 3. If the partial uotiets b0 b b... of α satisfy the growth restrictio o ( b = e ) as (for example b bouded or a polyomial i or b < e l ) of ( the µα ( )=. If b = e ) as (for istace b < e ( 3 ) ) the βα ( )=. Proof. By (7) (8) we have + 0 µα ( ) limsup l l bf 0 + l βα ( ) limsup l b. F Sice l F = l φ O( ) as (see Example 3) the first statemet follows. The secod statemet is immediate (ote that 3< φ ). 4. PRESCRIBING IRRATIONALITY MEASURES VIA JARNÍK'S THEOREM I 93 V. Jarík proved a geeral result o simultaeous Diophatie approximatio of s umbers [3 Satz 6] (see also [ p. 8 Exercise.5]). We recall the statemet for s = ad give his costructio by cotiued fractios (Theorem ). We the poit out that Jarík's method produces umbers with prescribed irratioality measure (Theorem 3 ad Corollary 4). I the ext sectio bypassig Jarík's method we explicitly costruct series with irratioality base ay prescribed real umber β >. As a preview of the two methods here are umbers with the same irratioality base but differet approximatio properties oe a cotiued fractio ad the other a series. Example 5. We have β( θ) = for the cotiued fractio θ = [ 037 ; b 5 b 6...] where b + is related to the -th coverget p by the formula (9) b 9 5 I particular b 5 = + 9 > 0. Proof. Takig ω ( x) + = +. = x i Corollary 4 below formula (9) satisfies coditios (0). Example 6. We have βτ ( ) = for the series
9 = + + +. τ Proof. Take β = i () below. Remark. Although the umbers i Examples 5 ad 6 have the same irratioality base βθ ( ) = βτ ( ) = they have opposite approximatio properties: for ay itegers p with > 0 sufficietly large we will see that p θ < τ p > I order to state Jarík's theorem we eed a defiitio. (We use the termiology of [ Chapter I 3].) Defiitio 4. (Jarík ) Give a real umber θ ad a real-valued fuctio ω ( x ) defied ad positive for x we say that θ is approximable to order ω if for every C > 0 there exists a pair of itegers p with > C such that p θ < ω( ). Theorem. (Jarík) Suppose that the fuctio ω ( x ) is defied for x positive ad decreasig ad that ω ( x) = o( x ) as x. The there exists a irratioal umber θ which is approximable to order ω but ot to order cω for ay costat 0< c <. Proof. (Sketch) Jarík costructs a cotiued fractio by choosig the partial uotiets b b... successively so that b + is related to the th coverget p by the formulas (0) b+ ω ( ) > lim b+ ω ( ) =. The he shows that ay umber θ = [ 0; b b...] which satisfies (0) also satisfies the coclusios of the theorem. The followig is a simple applicatio of Jarík's theorem. Theorem 3. Suppose that f( x λ ) is a irratioality measure with a fixed value of λ > 0 such that f( x λ ) = o( x ) as x. Suppose further that for ay ε > 0 there exists a positive costat c = c( ε) < idepedet of x such that f ( x λ + ε) < cf ( x λ) for x sufficietly large. The there exists a irratioal umber θ λ which has irratioality measure f( x λ ) ad is approximable to order ω( x) = f( x λ)..
0 Remark. Although a irratioality measure is strictly decreasig i both variables the secod hypothesis is ot redudat because c does ot deped o x. Proof. The fuctio ω( x) = f( x λ) satisfies the hypotheses of Theorem. Let θ λ be a umber satisfyig the coclusios. Sice f( x λ ) ad thus ω ( x ) is a decreasig fuctio of x ad θ λ approximable to order ω it follows that λ( θλ ) λ (i the otatio of Defiitio ). To show that λθ ( λ ) λ suppose o the cotrary that λ( θλ )> λ. The there exist ε > 0 ad ifiitely may itegers p with > 0 such that θλ p f( λ + ε) < cf( λ) = cω( ) for some costat 0< c <. But the θ λ is approximable to order cω which is a cotradictio. Hece λθ ( )= λ ad the result follows. λ Corollary 4. Every real umber β > is the irratioality base of some umber θ β which is approximable to order β x ; ay cotiued fractio θ β = [ 0; b b...] satisfyig (0) x with ω( x) = β will do. The aalogous result for µ > ad the irratioality expoet µ holds with ω ( x) = x. Proof. This follows from Theorem 3 ad the proof of Theorem. 5. PRESCRIBING IRRATIONALITY BASES VIA SERIES base. By a differet method we prove a related existece result for the irratioality Theorem 4. Ay real umber β > ca be realized as the irratioality base of a explicitly costructed series τ β which is ot approximable to order β x ; i fact () τ β p > for all itegers p with > 0 sufficietly large. β If β = ba> is a ratioal umber i lowest terms with b > 0 the b b b. () τβ = β + β + β + To prove Theorem 4 we first prove a lemma i which we costruct a series with prescribed irratioality base from a seuece of ratioal umbers with certai properties. We the prove the theorem by costructig the reuired seuece. The costructios per se do ot ivolve cotiued fractios but the proof of the lemma does.
Lemma 3. Give a real umber β > suppose that r a = b is a seuece of reduced fractios with positive deomiators satisfyig the coditios (3) > r r r3 β (4) lim r = β (5) b b + ad (6) b+ t where t is the tower of height defied by (7) t 0 = t = b = b b t b.... The the sum of the series (8) τ β = r = t b b b a = + a + a b b 3 b 3 + has irratioality base β. Moreover τ β satisfies (). Proof. Coditios (3) ad (7) imply that the series coverges. Sice ( a b ) = for coditios (5) ad (7) imply that the th partial sum of the series has deomiator t so that for some iteger u t a i i u (9) s := ri = = t t i= i= t i i ( u t ) =. From (3) (7) ad (8) it follows that there is a costat C > such that t t + + (0) r < τ β s < Cr. Usig (9) (0) ad (4) we deduce that if 0< ε < β the
u () 0 < τ β < t t < β ε t ( ) ( ε ). It follows that τ β is irratioal ad that βτ ( β ) β. To show that βτ ( β ) β we prove the stroger coditio (). By Lemma it suffices to show that the covergets pm m of τ β satisfy (). By (9) () ad Lemma all but a fiite umber of partial sums are covergets () s where pm = 0 m (3) m = t. Usig (3) () (0) ad (3) we have pm t = < τ < Cr t β 0. m+ m m+ m Sice 0< r < there exists such that if the 0 m + t t r log( t ) > >. Ct logβ Now cosider a coverget pm m of τ β with m large. If pm m is a partial sum () the (3) ad the lower boud i (0) imply that pm m satisfies ieuality (). If pm m is ot a partial sum the from () we have m + m< m+ for some = ( m). Usig (6) ad (3) we obtai log t ( b + ) < m m m t + < + = + logβ Hece for m ad thus sufficietly large.
3 τ β p p + p p m m m m + τβ > Cr m m + m m+ m+ m t + > = t > t t b β + + + m so that p m m satisfies ieuality (). This completes the proof of the lemma. Proof of Theorem 4. Give β > costruct a seuece of ratioal umbers r as follows. Case. If β is ratioal say β = ba with 0 < a< b ad ( ab ) = take the costat seuece r = a b for. Case. If β is irratioal express β i "base " as = k β = with 0 < k < k < k 3 <. Write the th partial sum of the series as S = A B where k A is odd ad B =. Choose a positive iteger such that B T where T deotes the tower of height defied by T 0 = T = B T ; for set r = S. Now choose > such that B T 3 where T = B T + ; for + set r = S. Cotiuig i this way we defie r for all. I both Cases ad the seuece r r r3... satisfies coditios (3) to (7) (i Case for (7) we have t = T ). The theorem ow follows from Lemma 3. ACKNOWLEDGEMENTS. I am grateful to Ya Bugeaud Yuri Nestereko Taguy Rivoal Michel Waldschmidt ad Wadim Zudili for valuable suggestios ad ecouragemet.
4 REFERENCES. Y. Bugeaud Approximatio by Algebraic Numbers Cambridge Uiversity Press Cambridge Eglad 004.. N. Fel'dma ad Y. Nestereko Number Theory IV: Trascedetal Numbers Spriger-Verlag Berli 998. 3. V. Jarík Über die simultae diophatische Approximatioe Math. Zeit. 33 (93) 505-543. 4. S. Lag Itroductio to Diophatie Approximatios Spriger-Verlag New York 995. 5. A. M. Legedre Théorie des Nombres 3rd ed. Tome Didot Paris 830; reprited by A. Blachard Paris 955. 6. J. Sodow Criteria for irratioality of Euler's costat Proc. Amer. Math. Soc. 3 (003) 3335-3344. 7. A faster product for π ad a ew itegral for l( π ) Amer. Math. Mothly to appear. 09 WEST 97TH STREET NEW YORK NY 05 USA email address: jsodow@alumi.priceto.edu