Simultaneous Approximation of G li d G ld R tiGeneralized Golden Ratios

Kari Frazer Lock

Williams College ‘04

Advisor: Edward Burger

Mathfest ‘04

Approximating ϕ

618034151 =+=ϕ ...618034.12

==ϕ

618.1500809

10001618

==500000

...61805.1233=

144

Best Approximate

DEFINITION. A reduced rational number, , is a b i l i i (b i )

QP

best rational approximation (best approximate) to an irrational number α, if for all satisfying

with we haveQq ≤≤1qp

Ppwith , we haveQq ≤≤1Qq

p≠

pPqpq

QPQ −<− αα

Continued Fraction Expansion

11

1

0

++=

aaα

11

1

3

2

1

++

+

aa

a

...1

54

3

++

aa

a

The Law of Best Approximates (Lagrange): The convergents are the complete set of The convergents are the complete set of best approximates!

Best Approximates

Nearest Integer Function:

|| x || = min { |x-k| : k ∈ Z }

|||| p|||| qα pq −= αqpq −= α

Denominators of best approximates are precisely the sequence of integers minimizing || αq ||.

Simultaneous Approximation

Let α0, α1, α2, …, αk be k+1 irrational numbers.

The best simultaneous approximates forThe best simultaneous approximates for these irrationals are the sequence of integers successively minimizingintegers successively minimizing

||}||||||||max{|| 10 qqq kααα ||}.||||,...,||||,max{|| 10 qqq kααα

Previous ResultsTHEOREM (Burger, 2000). Let α, α1, …, αK ∈ Q + Qαbe K + 1 irrational numbers. Then the sequence Dqj

*, Dqj+1*, Dqj+2

*, … is the complete list of values for integers q > L that successively minimize the

titquantity

||}||||,...,||||,max{|| 1 qqq kααα

where qj* denotes the denominator of the jth

t f th ili b *

||}||||, ,||||,{|| 1 qqq k

convergent of the auxiliary number α*.

Generalized Golden Ratios

24

11 2 ++

=+=aaaaϕ2

11

++

+

aa

a

...++

aa

Generalized Fibonacci Numbers:q0 = 1, q1 = aq0 1, q1 aqi = aqi-1 + qi-2

Generalized Golden Ratios

42 ++=

aaϕ ( )dQ∈2aϕ ( )Q

L if d l if th i t( )dQϕLEMMA. if and only if there exists an integer solution, t, to the Pell equation

( )da Q∈ϕ

422 −=− dta

THEOREM

THEOREM. Let (a0, t0), (a1, t1), … be the sequence giving all positive integral solutions topositive integral solutions to

)1(422 −=− dtain order of increasing magnitude. This sequence isin order of increasing magnitude. This sequence is nonempty (and infinite) if and only if the continued fraction expansion for has odd period length, and moreover, if ( t ) i th i i iti l ti t (1) th ll th

d(a0, t0) is the minimum positive solution to (1), then all other solutions are defined recursively by

( ))2()2(1)( 22 ++++ ttttdt ( ).)2(,)2(2

),( 1001010010 −−−− ++++= nnnnnn atatattdaaata

THEOREM

THEOREM. Let and let

where is the minimum generalized golden ratio in

( ),Q,...,,10

dkaaa ∈ϕϕϕ][aa =ϕ

ϕwhere is the minimum generalized golden ratio in this quadratic field. If then the

li d Fib i b i t d ith

0aϕ

,44

44

20

2

20

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

++

+++

=aa

aam kk

generalized Fibonacci numbers associated with a0,

greater than m, form the complete sequence of integers

q > m that successively minimize

||}||||||||max{|| qqq ϕϕϕ ||}.||||,...,||||,max{||10

qqqkaaa ϕϕϕ

)5(Q∈aϕ

ϕ ϕ4 ϕ11

1 1 4 111 1 4 11

2 3/2 8/2 22/2

3 5/3 13/3 33/3

5 8/5 21/5 55/55 8/5 21/5 55/5

8 13/8 34/8 89/8

13 21/13 55/13 144/1313 21/13 55/13 144/13

Thank you! ☺

kKari Frazer Lock

karisk8s@hotmail.com

Advisor: Edward BurgerAdvisor: Edward Burger

Best Approximates to ϕ

22= ...61904.134

=618=

5.13

21

=

=

...61764.13455

...61904.121

=625.113

6.15

=

=

...66666.135

5.12

= ...61818.1558934

=...61538.113218

=3 5513

618034151+ϕ ...618034.12

==ϕ