3 Gauss map and continued fractions

Transcription

1 ICTP, Trieste, July 08 Gauss map ad cotiued fractios I this lecture we will itroduce the Gauss map, which is very importat for its coectio with cotiued fractios i umber theory. The Gauss map G : [0, ] [0, ] is the followig map: { 0 if 0 G(x) = { } x mod if 0 < x Here {x} deotes the fractioal part of x. We ca write {x} = x [x] where [x] is the iteger part. Equivaletly, {x} = x mod. Remark that [ ] = x x < + + < x. Thus, explicitely, oe has the followig expressio (see the graph i Figure ): { 0 if 0 G(x) = x if + < x for N. The rescritio of G to a iterval of the form (/ +, /] is called brach. Each brach G : (/ +, /] [0, ) is mootoe, surjective (oto [0, )) ad ivertible (see Figure ) /4/ / Figure : The first braches of the graph of the Gauss map. The Gauss map is importat for its coectios with cotiued fractios. A fiite cotiued fractio (CF will be used as shorteig for Cotiued Fractio) is a expressio of the form () a 0 + a + a + a +... a where a 0, a, a,..., a N\{0} are called etries of the cotiued fractio expasio. We will deote the fiite cotiued fractio expasio by [a 0, a, a,..., a ]. Every fiite cotiued fractio expasio correspod to a ratioal umber p/q (which ca be obtaied by clearig out deomiators).

2 ICTP, Trieste, July 08 Example.. For example + = + = 7. Coversely, all ratioal umbers i [0, ] admit a represetatio as a fiite cotiued fractio. Example.. For example 4 = +, = Every irratioal umber x (0, ) ca be expressed through a (uique) ifiite cotiued fractio, that we deote by Example.. For example [a 0, a, a, a,... ] = a 0 + a + a + a +... π = +, = The umber ( 5 )/ is kow as golde mea ad it has the lowest possible cotiued fractio etries, all etries equal to oe. Similarly, the umber whose CF etries are all equal to is kow as silver mea. Oe ca see that a umber is ratioal if ad oly if the cotiued fractio expasio is fiite. If x is a irratioal umber whose ifiite cotiued fractio expasio is [a, a, a,... ], oe ca trucate the cotiued fractio expasio at level ad obtai a ratioal umber that we deote p /q p q = [a 0, a, a,..., a ]. These umbers p /q are called covergets of the cotiued fractio. Two of the importat properties of covergets are the followig:. Oe ca prove that p /q coverge to x expoetially fast, i.e. p lim = x ad p q x q ( ). () Thus, the fractios p /q give ratioal approximatios of x. This represetati is ot uique: if the last digit a of a fiite CF is, the [a 0,..., a, ] = [a 0,..., a + ]. If oe requires that the last etry is differet tha oe, though, the oe ca prove that the represetatio as fiite cotiued fractio is uique. To be precise, whe we write such a ifiite cotiued fractio expressio, its value is the limit of the fiite cotiued fractio expasio trucatios [a 0, a, a, a,..., a ], each of which is a well defied ratioal umber. Oe should first prove that this limit exist, see (). 5+, kow as golde ratio. It appears ofte i art ad i ature The iverse of the golde mea is sice it is cosidered aesthetically pleasig: for example, the ratio of the width ad height of the facade of the Parteo i Athes is exactly the golde ratio ad a whole Reessaice treaty, Luca Pacioli s De divia proportioe, writte i 509, is dedicated to the golde ratio i arts, sciece ad architecture...

3 ICTP, Trieste, July 08. Covergets give best approximatios amog all ratioal approximatios with deomiator up to q, that is x p x p q, p Z, 0 q q. q Oe ca also see that the cotiued fractio expasio of a irratioal umber is uique. To fid the cotiued fractio expasio of a umber, we will exploit the relatio with the symbolic codig of the Gauss map, i the same way that biary expasios are related to the symbolic codig of the doublig map. Let P be the subitervals of [0, ) aturally determied by the domais of the braches of the Gauss map: ( ] P =,, P = (, ] (, P = 4, ] (,..., P = +, ],... Remark that P accumulate towards 0 as icreases If we add P 0 = {0}, the collectio {P 0, P,..., P,... } is a (coutable) partitio 4 of [0, ]. Theorem.. Let x be irratioal. Let a 0, a,..., a,... be the itierary of O + G (x) with respect to the partitio {P 0, P, P,..., P,... }, i.e. x P a0, G(x) P a,..., G (x) P a,..., G k (x) P ak,..., The [a 0, a, a,..., a,... ]. Thus, itieraries of the Guass map give the etries of the cotiued fractio expasios. Remark.. If x is ratioal, the there exists such that G (x) = 0 ad hece G m (x) = 0 for all m. I this case, G m (x) P 0 for all m so the itierary is evetually zero. The theorem is still true if we cosider the begiig of the itieary: the fiite itierary before the tail of 0 gives the etries of the fiite cotiued fractio expasio of x. Proof. Let us first remark that x P + < x [ ] x < + =. () x I particular, a 0 = [/x] sice x P a0. Thus, { } G(x) = = [ ] x x = x x a 0 a 0 + G(x). Let us prove by iductio that [ ] a = G ad (x) a 0 + a +... a+g + (x) = [a 0, a,..., a + G + (x)]. (4) We have already show that this is true for = 0. Assume that it is proved [ for ] ad cosider +. Sice G + (x) P a+ by defiitio of itierary, we have a + = by (). This G (x) proves the first part of (4) for +. The, recallig the defiitio of G we have [ ] G + (x) = G + (x) G + = (x) G + (x) a + G + (x) = 4 Recall that a partitio is a collectio of disjoit sets whose uio is the whole space. a + + G + (x)

4 ICTP, Trieste, July 08 so that, pluggig that i the secod part of the iductive assumptio (4) we get a a +G + (x) =, a a + a + +G + (x) which proves the secod part of (4) for +. Thus, recursively, the itierary is producig 5 the ifiite cotiued fractio expasio of x. From the proof of the previous theorem, oe ca see the followig. Remark.. The Gauss map acts o the digits of the CF expasio as the oe-sided shift, that is if [a 0, a, a,..., a,... ] the G(x) = [a, a, a,..., a +,... ]. Oe ca characterize i terms of orbits of the Gauss map various class of umbers. For example:. Ratioal umbers are exactly the umbers x which have fiite cotiued fractio expasio or equivaletly such that there exists N such that G (x) = 0 (evetually mapped to zero by the Gauss map).. Quadratic irratioals, that is umbers of the form a+b c d, where a, b, c, d are itegers 6, are exactly umbers which have a evetually periodic cotiued fractio expasio or equivaletly are pre-periodic poits for the Gauss map. I umber theory (ad i particular i Diophatie approximatio) other class of umbers (for example Badly approximable umbers) ca be characterized i terms of their cotiued fractio expasio 7. Example.4. We have already see two examples of quadratic irratioals, the golde mea g ad the silver mea s: g = =, s = =. Both the golde mea ad the silver mea are fixed poits of the Gauss map: G(g) = g, G(s) = s. Similarly all other fixed poits correspod to umbers whose cotiued fractio etries are all equal. Example.5. Let α = + 5. The oe ca check that α = [,,,,,,... ], so that the etries are periodic ad the period is. Thus G (α) = α. Explicitely, sice we kow the itierary of α, we ca write dow the equatio satisfied by α. We kow that G(α) = α, sice [ α ] =, ad G(G(α)) = G(α) sice [ ] G(α) 5 Oe should still prove that the fiite cotiued fractios i (4) do coverge, as teds to ifiity ad that the limit is x. This ca be doe by the same method that oe ca use to show that covergets ted to x expoetially fast. 6 Equivaletly, oe ca defie quadratic irratioals as solutios of equatios of degree two with iteger coefficiets. 7 Oe ca defied Badly approximable umbers as the umbers for which there exists a umber A such that all etries a of their cotiued fractio expasio are bouded by A. I particular, quadratic irratioals are badly approximable. =, 4

5 ICTP, Trieste, July 08 so that the equatio G (α) = α becomes α = α. Usig the ideas i the previous exercise, oe ca produce quadratic irratioals with ay give periodic sequece of CF etries. Exercise.. Prove that if G (x) = 0 the x has a represetatio as a fiite cotiued fractio expasio ad thus it is ratioal. Exercise.. Prove that if G (x) = x the x satisfies a equatio of degree two with iteger etries. Coclude that x is a quadratic irratioal. 5