even integer continued fractions: a geometric approach · contents 1 introduction simple continued...

$Page 1: Even integer continued fractions: A geometric approach · Contents 1 Introduction Simple continued fractions Even integer continued fractions 2 An underlying geometry The theta graph$

Even integer continued fractions: A geometricapproach

Mairi Walker

The Open [email protected]

2nd July 2014

Mairi Walker (The Open University) Even integer continued fractions 2nd July 2014 1 / 25

Contents

1 IntroductionSimple continued fractionsEven integer continued fractions

2 An underlying geometryThe theta graphPaths in the theta graph

3 A geometric approachThe basicsSome classical results

Contents

Introduction Simple continued fractions

Simple continued fractions

A simple continued fraction is a continued fraction of the form

[b1,b2, . . . ,bn] = b1 +1

b2 +1

b3 + · · ·+ 1bn

,

where b1 ∈ Z and b2,b3, . . . ,bn are positive integers.

An infinite simple continued fraction is defined to be the limit

[b1,b2, . . . ] = limi→∞

[b1,b2, . . . ,bi ],

of its sequence of convergents, if it exists.

Properties of simple continued fractions

Simple continued fractions have some useful properties:

• Every infinite simple continued fraction converges.

• Every real number x has a unique simple continued fractionexpansion, which is finite if and only if x ∈ Q.

• The unique simple continued fraction expansion of a real numberis given by the Euclidean algorithm.

Plus plenty of deeper and more surprising properties...

General integer continued fractions do not satisfy such nice properties!

Introduction Even integer continued fractions

Even integer continued fractions

DefinitionA finite even integer continued fraction is a continued fraction of theform

[b1,b2, . . . ,bn] = b1 +1

b2 +1

b3 + · · ·+ 1bn

,

where b1 ∈ 2Z and b2,b3, . . . ,bn ∈ 2Z \ {0}.

An infinite even integer continued fraction is defined to be the limit

[b1,b2, . . . ] = limi→∞

[b1,b2, . . . ,bi ],

of its sequence of convergents, if it exists.

Introduction Even integer continued fractions

Properties of even integer continued fractions

Even integer continued fractions do actually have some usefulproperties:

• Every infinite even integer continued fraction converges.

• Every real number x has an even integer continued fractionexpansion, which is finite if and only if x = p

q is a reduced rationalwith p + q ≡ 1 (mod 2), and unique unless x = p

q is a reducedrational with p + q ≡ 0 (mod 2).

• The even integer continued fraction expansion of a real number isgiven by the nearest-integer algorithm.

An underlying geometry The theta graph

An underlying geometryLet S(z) = z + 1 and T (z) = −1

z . Then

[2,4,−2] = 2 +− 1

4 +− 1−2

= S2T(

4 +−1−2

)= S2TS4T (−2)

= S2TS4TS−2(0)

In general[b1,b2, . . . ,bn] = Sb1TSb2T . . .Sbn (0).

z . Then

[2,4,−2] = 2 +− 1

4 +− 1−2

= S2T(

4 +−1−2

)

= S2TS4T (−2)

= S2TS4TS−2(0)

z . Then

[2,4,−2] = 2 +− 1

4 +− 1−2

= S2T(

4 +−1−2

)= S2TS4T (−2)

= S2TS4TS−2(0)

z . Then

[2,4,−2] = 2 +− 1

4 +− 1−2

= S2T(

4 +−1−2

)= S2TS4T (−2)

= S2TS4TS−2(0)

z . Then

[2,4,−2] = 2 +− 1

4 +− 1−2

= S2T(

4 +−1−2

)= S2TS4T (−2)

= S2TS4TS−2(0)

The theta group

The elements S2 and T generate a group called the theta group, Γθ,which is the group of Möbius maps of the form

f (z) =az + bcz + d

where a,b, c,d ∈ Z, ad − bc = 1 and(a bc d

)≡

(1 00 1

)or

(a bc d

)≡

(0 11 0

)(mod 2).

Elements of the theta group are of the form

Sb1TSb2T . . .Sbn

where bi ∈ 2Z, and bi 6= 0 for i = 2, . . . ,n − 1.

The theta group

The elements S2 and T generate a group called the theta group, Γθ,which is the group of Möbius maps of the form

f (z) =az + bcz + d

where a,b, c,d ∈ Z, ad − bc = 1 and(a bc d

)≡

(1 00 1

)or

(a bc d

)≡

(0 11 0

)(mod 2).

Elements of the theta group are of the form

Sb1TSb2T . . .Sbn

where bi ∈ 2Z, and bi 6= 0 for i = 2, . . . ,n − 1.

The modular group

The theta group is closely related to the modular group, Γ, which is thegroup generated by S and T . The modular group is the group ofMöbius maps of the form

f (z) =az + bcz + d

where a,b, c,d ∈ Z and ad − bc = 1.

The modular group is a Fuchsian group: A discrete group of isometriesof the hyperbolic upper half-plane.

The theta group is an index 3 normal subgroup of the modular group,so it is also a Fuchsian group.

The theta group

In fact, the maps S2 and T act on the complex plane as follows:

2

Let L denote the hyperbolic geodesic joining 0 to∞ in the upperhalf-plane. Then the orbit of L under elements of the theta group willform a tessellation of the upper half-plane.

The theta graph

The theta graph, Gθ, is the orbit of L under elements of the theta group.

The graph F∞

1

It is a connected planar graph.

The theta graph

The graph F∞

1

• Gθ is a tree.

• Vertices of Gθ are the reduced rationals pq with p + q ≡ 1 (mod 2),

plus∞.

Vertices surrounding a ‘face’ accumulate at a face-centre.

• Face-centres are the reduced rationals pq with p + q ≡ 0 (mod 2).

The theta graph

The graph F∞

1

• Gθ is a tree.• Vertices of Gθ are the reduced rationals p

q with p + q ≡ 1 (mod 2),plus∞.

The theta graph

The graph F∞

1

The theta graph

The graph F∞

1

An underlying geometry Paths in the theta graph

Paths in the theta graph

Recall that[b1,b2, . . . ,bn]q = Sb1TSb2T . . .TSb2(0),

so the vertices of Gθ are exactly the numbers that can be representedby finite even integer continued fractions.

The convergents of an even integer continued fraction [b1,b2, . . . ,bn]qare the numbers Ck = [b1,b2, . . . ,bk ]q for each k = 1,2, . . . ,n − 1.Each convergent is therefore also a vertex of Gθ.

TheoremA sequence of vertices∞ = v1, v2, . . . , vn = x form a simple path in Gθ

if and only if they are the consecutive convergents of an even integercontinued fraction expansion of x.

C1 = [0], C2 = [0,−2], C3 = [0,−2,−2], C4 = [0,−2,−2,2], . . .

A geometric approach The basics

Convergence and uniquenessIt is immediate that there is a unique finite even integer continuedfraction expansion of any reduced rational number p

q with p + q ≡ 1(mod 2).

The rest of the path is bounded between the pairs of face-centres.These intervals shrink to zero, giving convergence.

Two continued fractions converging to the same limit must eventuallylie in the same face-centre interval... unless the limit is a face-centre.

Convergence and uniqueness

In fact, since the intervals defined by face-centres form partitions, theycan be chosen to converge to any real number.

This gives us the following result:

TheoremEvery real number x has an even integer continued fraction expansionwhich is finite if and only if x = p

q is a reduced rational with p + q ≡ 1(mod 2), and unique unless x = p

q is a reduced rational with p + q ≡ 0(mod 2).

These properties are very similar to those for simple continuedfractions.

Existence

Join x to∞ by a geodesic.

Existence

Choose the unique face F1 incident to∞ that intersects this geodesic.Mark the face-centre c1 of this face.

Existence

If x ≤ c1 travel anti-clockwise around the face. If x > c1 travelclockwise. If x = c1 we’re done.

Existence

If not, continue until you reach a vertex of the unique face F2 adjacentto F1 intersecting the geodesic.

If x ≤ c2 travel anti-clockwise around the face. If x > c2 travelclockwise. If x = c2 we’re done...

A geometric approach Some classical results

A theorem of Serret

A tail of the continued fraction [b1,b2, . . . ] is [bi ,bi+1, . . . ] for some i .

Two numbers are equivalent if there is a function in the extendedmodular group mapping one to the other.

Theorem (Serret)Two irrational numbers are equivalent if and only if their simplecontinued fraction expansions have the same tail.

Two numbers are theta-equivalent if there is a function in the extendedtheta group mapping one to the other. Serret’s theorem holds:

TheoremTwo irrational numbers are theta-equivalent if and only if their eveninteger continued fraction expansions have the same tail.

A theorem of Serret

Two numbers are theta-equivalent if there is a function in the extendedtheta group mapping one to the other.

Serret’s theorem holds:

A theorem of Serret

A theorem of Lagrange

A quadratic irrational is an irrational root of a quadratic equation. Theyare closely connected to periodic continued fractions, those of the form

[b1,b2, . . . ,bk ,bk+1,bk+2, . . . ,bk+t ].

Theorem (Lagrange)A real number x has a periodic simple continued fraction expansion ifand only if x is a quadratic irrational.

An analogous theorem holds for even integer continued fractions.

TheoremA real number x has a periodic even integer continued fractionexpansion if and only if x is a quadratic irrational or x is a face-centre.

[b1,b2, . . . ,bk ,bk+1,bk+2, . . . ,bk+t ].

Summary Summary

Summary

To summarise:• Even integer continued fractions have attractive properties

analogous to those for simple continued fractions.

• It’s often easier and more natural to study even integer continuedfractions as paths in Γθ.

Where next?• Diophantine approximation.• Gaussian integer continued fractions.

Summary Summary

Summary

analogous to those for simple continued fractions.• It’s often easier and more natural to study even integer continued

fractions as paths in Γθ.

Summary Summary

Summary

Where next?• Diophantine approximation.

• Gaussian integer continued fractions.

Summary Summary

Summary

Summary Summary

Thanks for listening!

:)

Bibliography

A.F. Beardon, M. Hockman, I. Short.Geodesic Continued Fractions.Michigan Mathematical Journal, 61(1):133–150, 2012.

C. Kraaikamp, A. Lopes.The Theta Group and the Continued Fraction Expansion with EvenPartial Quotients.Geometriae Dedicata, 59:293–333, 1996.

J. H. Conway.The Sensual (Quadratic) Form.1997

even integer continued fractions: a geometric approach · contents 1 introduction simple continued...

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