Gaussian Integer Continued Fractions · Integer continued fractions The geometry of integer...

Gaussian Integer Continued Fractions

Mairi Walker

The Open [email protected]

Young Researchers in Mathematics Conference

17th August 2015

Mairi Walker (The Open University) Gaussian Integer Continued Fractions 17th August 2015 1 / 22

Contents

1 Integer continued fractionsInteger continued fractions and the modular groupThe geometry of integer continued fractions

2 Gaussian integer continued fractionsGaussian integer continued fractions and the Picard groupThe geometry of Gaussian integer continued fractions

3 A geometric approach to Gaussian integer continued fractionsThe geometry of Gaussian integer continued fractionsConvergence

Integer continued fractions Integer continued fractions and the modular group

Integer continued fractions

DefinitionAn integer continued fraction is an expression of the form

[b1,b2,b3, . . . ] = b1 −1

b2 −1

b3 − . . .

where bi ∈ Z for i = 1,2, . . . ,n.

The convergents of an integer continued fraction are the values of thefinite integer continued fractions [b1,b2, . . . ,bk ] for k = 1,2, . . . .

If the sequence of convergents converges to a real number x , we saythat the integer continued fraction converges to x , or is an expansionof x .

Integer continued fractions

DefinitionAn integer continued fraction is an expression of the form

[b1,b2,b3, . . . ] = b1 −1

b2 −1

b3 − . . .

where bi ∈ Z for i = 1,2, . . . ,n.

The convergents of an integer continued fraction are the values of thefinite integer continued fractions [b1,b2, . . . ,bk ] for k = 1,2, . . . .

If the sequence of convergents converges to a real number x , we saythat the integer continued fraction converges to x , or is an expansionof x .

The modular groupThe modular group, Γ, is the group generated by the Möbiustransformations

S(z) = z + 1 and T (z) = −1z.

The elements of Γ are the Möbius transformations

f (z) =az + bcz + d

with a,b, c,d ∈ Z and ad − bc = 1. They form a discrete group ofisometries of the hyperbolic upper half-plane H.

S(z) = z + 1 and T (z) = −1z.

f (z) =az + bcz + d

with a,b, c,d ∈ Z and ad − bc = 1. They form a discrete group ofisometries of the hyperbolic upper half-plane H.

S(z) = z + 1 and T (z) = −1z.

f (z) =az + bcz + d

with a,b, c,d ∈ Z and ad − bc = 1. They form a discrete group ofisometries of the hyperbolic upper half-plane H.Mairi Walker (The Open University) Gaussian Integer Continued Fractions 17th August 2015 4 / 22

The modular group and continued fractions

Any element of Γ can be written Sb1TSb2T . . .SbnTSbn+1 .

Sb1TSb2T . . .SbnTSbn+1(∞) = b1 −1

b2 −1

b3 − · · · −1bn

Conversely, the word Sb1TSb2T . . .SbnT can be associated to[b1,b2, . . . ,bn].

There is a correspondence between elements of Γ and finite integercontinued fractions.

b2 −1

b3 − · · · −1bn

b2 −1

b3 − · · · −1bn

b2 −1

b3 − · · · −1bn

Integer continued fractions The geometry of integer continued fractions

The Farey graph

We work in the hyperbolic upper half-plane H. Let L denote the linesegment joining 0 to∞.

The Farey graph, F , is formed as the orbit of L under Γ: Edges areimages of L, and vertices are images of∞, under elements of Γ.

The Farey graph

F is the 1-skeleton of a tessellation of H by ideal hyperbolic triangles.

Hecke graphs in the plane

Figure 1: q = 3

Vertices lie on R ∪ {∞}. They are precisely the rational numbers, plus∞. Each vertex has infinite valency. The vertices neighbouring∞ arethe integers.

Paths in the Farey graph

[b1,b2, . . . ,bn] = Sb1TSb2T . . .SbnT (∞)

so finite integer continued fractions are vertices of F . In particular,each convergent of an infinite integer continued fraction is a vertex ofF .

TheoremA sequence of vertices∞ = v1, v2, v3 . . . forms an infinite path in F ifand only if they are the consecutive convergents of an infinite integercontinued fraction expansion.

There is a correspondence between integer continued fractions andpaths in the Farey graph with initial vertex∞.

[b1,b2, . . . ,bn] = Sb1TSb2T . . .SbnT (∞)

Take, for example, [0,−2,1,3, . . . ]

[0] = 0, [0,−2] =12, [0,−2,1] =

13, [0,−2,1,3] =

27, . . .

The geometry of integer continued fractionsWe can reformulate questions about continued fractions into questionsabout paths. This allows us to• Interpret the elementary theory of simple continued fractions

geometrically.

• Study the convergence of integer continued fractions.

• Investigate the approximation of irrational numbers by rationals.

Conversely, we can reformulate questions about paths into questionsabout continued fractions. This allows us to• Define the notion of a geodesic continued fraction.

• Characterise and enumerate geodesic continued fractions. (Seethe work of Beardon, Hockman and Short [1], for example.)

Can we develop a similar theory for other classes of continuedfractions?

geometrically.

Gaussian integer continued fractions Gaussian integer continued fractions and the Picard group

Gaussian integer continued fractionsDefinitionA Gaussian integer continued fraction is an expression of the form

[b1,b2,b3, . . . ] = b1 +1

b3 + . . .

where bi ∈ Z[i] = {x + iy | x , y ∈ Z} for i = 1,2, . . . ,n.

The convergents of a Gaussian integer continued fraction are thevalues of the finite Gaussian integer continued fractions [b1,b2, . . . ,bk ]for k = 1,2, . . . .

If the sequence of convergents converges to a real number x , we saythat the Gaussian integer continued fraction converges to x , or is anexpansion of x .

Gaussian integer continued fractionsDefinitionA Gaussian integer continued fraction is an expression of the form

[b1,b2,b3, . . . ] = b1 +1

b3 + . . .

where bi ∈ Z[i] = {x + iy | x , y ∈ Z} for i = 1,2, . . . ,n.

The convergents of a Gaussian integer continued fraction are thevalues of the finite Gaussian integer continued fractions [b1,b2, . . . ,bk ]for k = 1,2, . . . .

If the sequence of convergents converges to a real number x , we saythat the Gaussian integer continued fraction converges to x , or is anexpansion of x .Mairi Walker (The Open University) Gaussian Integer Continued Fractions 17th August 2015 11 / 22

The Picard groupThe Picard group, P, is the group generated by Möbius transformations

R(z) = z + i and S(z) = z +1 and T (z) = −1z

and U(z) =1z.

The elements of P are the Möbius transfromations

z 7→ az + bcz + d

with a,b, c,d ∈ Z[i] and |ad − bc| = 1. Their action can be extendedvia the Poincaré extension to isometries of the hyperbolic upperhalf-space H3, and again they form a discrete group.

and U(z) =1z.

z 7→ az + bcz + d

with a,b, c,d ∈ Z[i] and |ad − bc| = 1. Their action can be extendedvia the Poincaré extension to isometries of the hyperbolic upperhalf-space H3, and again they form a discrete group.

and U(z) =1z.

z 7→ az + bcz + d

with a,b, c,d ∈ Z[i] and |ad − bc| = 1. Their action can be extendedvia the Poincaré extension to isometries of the hyperbolic upperhalf-space H3, and again they form a discrete group.Mairi Walker (The Open University) Gaussian Integer Continued Fractions 17th August 2015 12 / 22

The Picard group and continued fractionsGiven α ∈ Z[i] we can define Sα ∈ P by Sα(z) = z + α.

Any element of P can be written Sb1USb2U . . .SbnUSbn+1 where eachbi ∈ Z[i].

Sb1USb2U . . .SbnUSbn+1(∞) = b1 +1

b3 + . . .1bn

Conversely, the word Sb1USb2U . . .USbn can be associated to[b1,b2, . . . ,bn].

There is a correspondence between elements of P and finite Gaussianinteger continued fractions.

b3 + . . .1bn

Gaussian integer continued fractions The geometry of Gaussian integer continued fractions

The Picard Graph

DefinitionThe Picard graph, G, is formed as the orbit of the vertical line segmentL with endpoints 0 and∞ under the Picard group.

It is a three-dimensional analogue of the Farey graph.

The Picard Graph

DefinitionThe Picard graph, G, is formed as the orbit of the vertical line segmentL with endpoints 0 and∞ under the Picard group.

It is a three-dimensional analogue of the Farey graph.

Properties of the Picard graph

• The Picard graph is the 1-skeleton of a tessellation of H3 by idealhyperbolic octahedra.

• The vertices V (G) are reduce quotients of Gaussian integers ac :

they are precisely those complex numbers with reduced rationalreal and complex parts, and∞.

• The edges of G are hyperbolic geodesics. Two vertices ac and b

dare joined by an edge in G if and only if |ad − bc| = 1.

Paths in the Picard graph

[b1,b2, . . . ,bn] = Sb1USb2U . . .USbnT (∞)

so finite Gaussian integer continued fractions are vertices of G. Inparticular, each convergent Ci of an infinite Gaussian integercontinued fraction is a vertex of G.

TheoremA sequence of vertices∞ = v1, v2, v3 . . . forms an infinite path in G ifand only if they are the consecutive convergents of an infiniteGaussian integer continued fraction expansion.

There is a correspondence between Gaussian integer continuedfractions and paths in the Picard graph with initial vertex∞.

A geometric approach The geometry of Gaussian integer continued fractions

The geometry of Gaussian integer continued fractionsWe can reformulate questions about continued fractions into questionsabout paths.• When does a Gaussian integer continued fraction converge?

• Under what conditions is there a unique expansion of everycomplex number? See, for example, Dani and Nogueira [2].

• Can we investigate the approximation of irrational complexnumbers by complex rational numbers?

Conversely, we can reformulate questions about paths into questionsabout continued fractions.• Can we define the notion of a geodesic Gaussian integer

continued fraction? (Yes.)

• Can we characterise and enumerate geodesic continuedfractions?

A geometric approach Convergence

Convergence of Gaussian integer continued fractions

Recall that a Gaussian integer continued fraction [b1,b2, . . . ]converges if its sequence of convergents [b1,b2, . . . ,bk ] converges. Doall Gaussian integer continued fractions converge?

[0, i , i , i , i , . . .] = 0 +1

i + . . .

[0] = 0, [0, i] = −i , [0, i , i] =∞, [0, i , i , i] = 0,

[0, i , i , i , i] = −i , [0, i , i , i , i , i] =∞, . . .

When does a Gaussian integer continued fraction converge?

[0, i , i , i , i , . . .] = 0 +1

i + . . .

[0] = 0, [0, i] = −i , [0, i , i] =∞, [0, i , i , i] = 0,

[0, i , i , i , i] = −i , [0, i , i , i , i , i] =∞, . . .

[0, i , i , i , i , . . .] = 0 +1

i + . . .

[0] = 0, [0, i] = −i , [0, i , i] =∞, [0, i , i , i] = 0,

[0, i , i , i , i] = −i , [0, i , i , i , i , i] =∞, . . .

Convergence of Gaussian integer continued fractionsLiterature on this topic generally restricts to certain classes ofGaussian integer continued fractions, such as those obtained usingalgorithms. See, for example, Dani and Nogueira [2].

Can we find a more general condition for convergence that can beapplied to all Gaussian integer continued fractions?

TheoremAn infinite path in G with vertices∞ = v1, v2, v3, . . . converges toz /∈ V (G) if and only if the sequence v1, v2, . . . contains no constantsubsequence and has only finitely many accumulation points.

CorollaryAn infinite Gaussian integer continued fraction converges to z /∈ V (G)if and only if its sequence of convergents contains no constantsubsequence and has only finitely many accumulation points.

Summary Summary

SummaryTo summarise:• Gaussian integer continued fractions can be viewed as paths in

the Picard graph.• This technique allows us to find and prove a simple condition for

the convergence of Gaussian integer continued fractions.

Where next?• Can we find conditions under which classical theorems of

continued fraction theory hold, expanding on the work of Dani andNogueira in [2]?

• What results can we obtain in the Diophantine approximation ofcomplex numbers? Can we expand on the work of Schmidt [3]?

• Can we use hyperbolic geometry to study other classes ofcomplex continued fractions?

Summary Summary

SummaryTo summarise:• Gaussian integer continued fractions can be viewed as paths in

the Picard graph.• This technique allows us to find and prove a simple condition for

the convergence of Gaussian integer continued fractions.

Where next?• Can we find conditions under which classical theorems of

continued fraction theory hold, expanding on the work of Dani andNogueira in [2]?

• What results can we obtain in the Diophantine approximation ofcomplex numbers? Can we expand on the work of Schmidt [3]?

• Can we use hyperbolic geometry to study other classes ofcomplex continued fractions?

Summary Summary

Thank you for your attention

Bibliography

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

S. Dani, A. Nogueira.Continued fractions for complex numbers and values of binaryquadratic forms.Transactions of the American Mathematical Society,366(7):3553–3583, 2014.

A. Schmidt.Diophantine approximation of complex numbers.Lecture Notes in Pure and Applied Mathematics, 92:353–377,1984.

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