An elementary proof that almost all real numbers are normal

Transcription

1 Acta Uiv. Sapietiae, Mathematica, 2, ( A elemetary proof that almost all real umbers are ormal Ferdiád Filip Departmet of Mathematics, Faculty of Educatio, J. Selye Uiversity, Rolícej šoly 59, 945 0, Komáro, Slovaia filip.ferdiad@selyeui.s Ja Šuste Departmet of Mathematics ad Istitute for Research ad Applicatios of Fuzzy Modelig, Faculty of Sciece, Uiversity of Ostrava, 30. duba 22, Ostrava, Czech Republic ja.suste@osu.cz Abstract. A real umber is called ormal if every bloc of digits i its expasio occurs with the same frequecy. A famous result of Borel is that almost every umber is ormal. Our paper presets a elemetary proof of that fact usig properties of a special class of fuctios. Itroductio The cocept of ormal umber was itroduced by Borel. A umber is called ormal if i its base b expasio every bloc of digits occurs with the same frequecy. More exact defiitio is Defiitio A real umber x (0, is called simply ormal to base b 2 if its base b expasio is 0.c c 2 c 3... ad #{ c = a} lim = b for every a {0,...,b }. 200 Mathematics Subject Classificatio: K6, 26A30 Key words ad phrases: ormal umber, measure 99

2 00 F. Filip, J. Šuste A umber is called ormal to base b if for every bloc of digits a...a L, L #{ L c + = a,...,c +L = a L } lim = b L. A umber is called absolutely ormal if it is ormal to every base b 2. A famous result of Borel [] is Theorem Almost every real umber is absolutely ormal. This theorem ca be proved i may ways. Some proofs use uiform distributio [5], combiatorics [7], probability [8] or ergodic theory [2]. There are also some elemetary proofs almost avoidig higher mathematics. Kac [3] proves the theorem for simply ormal umbers to base 2 usig Rademacher fuctios ad Beppo Levi s Theorem. illse [6] also cosiders biary case. He uses series of itegrals of step fuctios ad avoids usage of measure theory i the proof by defiig a ull set i a differet way. Khoshevisa [4] maes a survey about ow results o ormal umbers ad their cosequeces i diverse areas i mathematics ad computer sciece. This paper presets aother elemetary proof of Theorem. Our proof is based o the fact that a bouded mootoe fuctio has fiite derivative i almost all poits. We also use the fact that a coutable uio of ull sets is a ull set. Here is a setch of the proof. We itroduce a special class of fuctios. I Sectio 2 we prove elemetary properties of the fuctios F. We prove boudedess ad mootoicity ad assumig that the derivative F (x exists i poit x we prove that the product (5 has fiite value. We deduce that the product (5 has fiite value for almost every x. I Sectio 3 we prove that every o-ormal umber belogs to some set P. We tae a particular fuctio F. We fiish the proof by showig that for elemets of P the product (5 does ot have fiite value. For the proof of Theorem it is obviously sufficiet to cosider oly umbers i the iterval (0,. Defiitio 2 Let b = {b } be a sequece of itegers b 2. Let ω = {ω } be a sequece of divisios of the iterval [0, ], Put ω = {f (c} b c=0, f (0 = 0, f (c < f (c +, f (b =. (c := f (c + f (c.

3 Almost all umbers are ormal 0 Fuctio F b,ω : [0, ] [0, correspodig to b ad ω is defied as follows. For x [0,, let c x = b ( be its {b } -Cator series. The We defie F b,ω ( =. F b,ω (x := f (c (c. The reaso for defiig F b,ω ( = is that the actual rage of F b,ω is [0, ]. This is proved i Lemma 2. 2 Properties of the fuctio F I this sectio we derive some basic properties of a geeral fuctio F b,ω. Lemma allows us to express a particular value F(x i terms of values of some other fuctio F. For defie b ( := {b ( }, ω( := {ω ( } ad { ( } by Moreover, for x = b ( := b +, ω ( := ω +, ( := +. c b x ( := (0, defie Lemma (Shift property We have F b,ω (x = f (c (c + Proof. A easy computatio yields F b,ω (x = f (c (c c + b(. (c F b (,ω (x (. (

4 02 F. Filip, J. Šuste = + = f (c (c (c f + (c + f (c (c + + (c + (c F b (,ω (x (. ( Lemma 2 (Rage For x [0, ] the value F b,ω (x [0, ]. Proof. First we prove that for every b, ω ad every x = F b,ω ( c b = c b f (c (c. (2 We will proceed by iductio ( o. For = we have F c b,ω b = f (c. For + we use Lemma. By the iductio assumptio we have Hece F b (,ω ( (x (. F b,ω (x = f (c + (c F b (,ω ( (x ( f (c + (c = f (c +. have ow we use (2 ad pass to the limit. For x = F b,ω (x = lim f (c (c. Lemma 3 (Mootoicity The fuctio F b,ω is odecreasig. c b we

5 Almost all umbers are ormal 03 Proof. Let 0 x < y < be two umbers with x = c b ad y = d b. We prove that F b,ω (x F b,ω (y. Let be the iteger such that c = d for ad c < d. The Lemmas ad 2 imply F b,ω (x = = f (c (c + f (c (c + f (c f (d (c F b (,ω (x ( ( (c (c + f (c + (d + f (d f (d (d = F b,ω (y. (c (d For ad c {0,...,b } defie f (c := f (b c. Put ω := {{f (c} b c=0 }. Lemma 4 (Symmetry For every x = Proof. We have c b we have F b,ω ( x = F b,ω (x. (3 (c = f (c + f (c = (b c. ow we will proceed by iductio. For = we have F b,ω ( x = f (b c = f (c = F b,ω ( c b = F b,ω (x.

6 04 F. Filip, J. Šuste ow suppose that (3 holds for, F b,ω ( ( e e b = F b,ω b for every possible sequece {e }. The usig Lemma we obtai for x = + c b that F b,ω ( x = F b,ω ( b c b + b ( b + c + b + = f (b c + (b c F b (,ω ( ( x ( = f (b c + (b c ( F b (,ω ((x( = f (c + + (c ( F b (,ω ((x( = ( f (c + (c F b (,ω ((x( = F b,ω (x. + b ( + c ( + b + Remar Oe ca prove that if max (c = 0 the F b,ω is cotiuous o the iterval [0, ]. Oe ca the exted Lemma 4 for every x c=0,...,b [0, ]. Lemma 5 (Differece For every ( F b,ω c b + c ( + c b F b,ω b = (c. (4 Proof. Deote the left-had side of (4 by LHS. The if c b 2 the LHS = = ( ( c + ( c (c F b (,ω F ( b b (,ω ( b (c (f (c + f (c = (c.

7 Almost all umbers are ormal 05 I the case c = b we apply the first case o the fuctio F b,ω, LHS = = ( ( F b,ω ( ( F b,ω b c b b c b (b c = + (c. b I the followig text we will use the symbol Θ (c := b (c. Lemma 6 (Derivative Let x = c b derivative F b,ω (x exists ad is fiite. The (0,. Suppose that the F b,ω (x = Θ (c. (5 I particular, this product has a fiite value. Proof. We have lim ( F b,ω ( c b c b c = lim b + c + ( c b F b,ω b + c + ( c (6 b b + c + b x b (7

8 06 F. Filip, J. Šuste + ( F b,ω x b = F b,ω (x lim c b c + c + b F b,ω (x b c ( c b F b,ω (x F b,ω b x c b b b + c + b x = F b,ω (x. (8 (9 Existece of F b,ω (x implies that limits of (8 ad of the secod fractio i (9 are equal to F b,ω (x. Hece the limit (6 exists ad is equal to F b,ω (x. I c the case that x = b we obtai that (6 = F b,ω (x immediately. O the other had, Lemma 5 implies that (6 = lim (c = Θ (c. b Corollary For almost every x [0, ] the derivative F b,ω (x exists ad is c fiite. I particular, for almost every x = b the product Θ (c exists ad is fiite (possibly zero. Proof. The fuctio F b,ω is bouded ad odecreasig, hece i almost all poits it has a fiite derivative. Accordig to Lemma 6 we obtai that the product (5 is fiite. 3 Mai result Our mai result is a proof of Theorem.

9 Almost all umbers are ormal 07 Proof. A umber x (0, is ot absolutely ormal if there exist b 2, c L ad a,...,a L {0,...,b } such that if x = b the lim if #{ L c +i = a i, i =,...,L} The there exists s {0,...,L } such that lim if < b L. #{ L, s (modl c +i = a i, i =,...,L} Hece for some ratioal β < Lb L < Lb L. #{ L, s (modl c +i = a i, i =,...,L} lim if β. (0 c Deote by R b,l,a,s,β the set of all x = satisfyig (0. The result of b the previous paragraph is that the set of ot absolutely ormal umbers is a subset of b L b=2 L= a,...,a L =0 s=0 β (0, Lb L Q R b,l,a,s,β. It is sufficiet to prove that every set R b,l,a,s,β has zero measure. The the set of ot absolutely ormal umbers is a subset of a coutable uio of ull sets, hece it is a ull set. ( Let b 2, L, a,...,a L {0,...,b }, s {0,...,L } ad β 0, Lb. Put A = a b L + a L 2 b L a L. Let The obviously, x = c s b = d b + b s d s+ b L R b,l,a,s,β. # { L, s (modl c +i = a i, i =,...,L } Hece lim if M #{s < M d = A} M { = # s < = lim if [ s L ] d = A}. # { s < [ ] s L d = A } ] [ s L

10 08 F. Filip, J. Šuste = lim if Lβ. [ s L ] #{ L, s (modl c +i = a i, i =,...,L } From this we obtai that R b,l,a,s,β P, where P = { x = s d b + b s d s+ #{s < d = A} } lim if b L Lβ. Thus it is sufficiet to prove that the set P has zero measure. Let α ( Lβ, b L. For t [0, ] defie ϕ α (t := t α( b L t α. b L The fuctio ϕ α is cotiuous with ϕ α (0 = 0, ϕ α ( = ad ϕ α( = αb L b L < 0. Hece there is T (0, with ϕ α(t =. For u (0, put ψ(u := ϕ u (T = T u( b L T u. b L The fuctio ψ is cotiuous ad decreasig with ψ(α =. Cosider the fuctio F b,ω correspodig to b = {b } with ad ω = {ω } with b = { b, if s, b L, if > s, b, if s, T (d =, if > s ad d = A, b L, if > s ad d A. b L T b L (b L We have, if s, Θ (d = T, if > s ad d = A,, if > s ad d A. b L T b L

11 Almost all umbers are ormal 09 ow Corollary implies that for almost every x = followig product exists ad is fiite s d b + b s d +s b Ls the Θ (d = lim T#{s< d=a}( b L T #{s< d=a} b L ( ( #{s < d = A}. = lim ψ ( ow suppose that x P. The ( #{s < d = A} ( #{s < d = A} lim supψ = ψ lim if ψ(lβ > ψ(α =, hece ( ( #{s < d = A} lim sup ψ =, cotradictig fiiteess of (. Thus the set P has zero measure. Acowledgmet This research was supported by the grats VEGA /0753/0. GAČR 20/07/09 ad Refereces [] É. Borel, Les probabilités déombrables et leurs applicatios arithmétiques, Supplemeto di red. circ. Mat. Palermo, 27 (909, [2] W. A. Coppel, umber Theory: A itroductio to mathematics, Spriger Sciece & Busiess, [3] M. Kac, Statistical idepedece i probability, aalysis ad umber theory, Carus Math. Moographs, o. 2, Wiley, ew Yor, 959. [4] D. Khoshevisa, ormal umbers are ormal, Clay Mathematics Istitute Aual Report 2006, p. 5 (cotiued o pp

12 0 F. Filip, J. Šuste [5] L. Kuipers, H. iederreiter, Uiform distributio of sequeces, Pure ad Applied Maths. o. 29, Wiley, ew Yor, 974. [6] R. illse, ormal umbers without measure theory, Am. Math. Mothly, 07 (2000, [7] T. Šalát, Reále čísla, Alfa, Bratislava, 982. [8] M. Švec, T. Šalát, T. eubru, Matematicá aalýza fucií reálej premeej, Alfa, Bratislava, 987. Received: December 9, 2009; Revised: April 7, 200