Simultaneous Approximation of GlidGldRtiGeneralized Golden ...kfl5/Lock_Mathfest_2004b.pdf ·...
14
Click here to load reader
Transcript of Simultaneous Approximation of GlidGldRtiGeneralized Golden ...kfl5/Lock_Mathfest_2004b.pdf ·...
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
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
