The local Hurwitz constant and Diophantine approximation on Hecke groups
Authors: Lehner J. 
Published: June 2010

Define the Hecke group by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript q Baseline equals mathematical left-angle Start 2 By 2 Matrix 1st Row 1st Column 1 2nd Column a m p semicolon lamda Subscript q Baseline 2nd Row 1st Column 0 2nd Column a m p semicolon 1 EndMatrix comma Start 2 By 2 Matrix 1st Row 1st Column 0 2nd Column a m p semicolon negative 1 2nd Row 1st Column 1 2nd Column a m p semicolon 0 EndMatrix mathematical right-angle comma">

G
q

=

⟨

(

1

a
m
p
;

λ
q

0

a
m
p
;
1

)

,

(

0

a
m
p
;

−
1

1

a
m
p
;
0

)

⟩

,

{G_q} = \left \langle {\left ( {\begin {array}{*{20}{c}} 1 & {{\lambda _q}} \\ 0 & 1 \\ \end {array} } \right ),\left ( {\begin {array}{*{20}{c}} 0 & { - 1} \\ 1 & 0 \\ \end {array} } \right )} \right \rangle ,

\]

λ
q

=
2
cos
⁡
π

/

q

{\lambda _q} = 2\cos \pi /q

,

q
=
3
,
4
,
…

q = 3,4, \ldots

. We call

G
q

(
∞
)

{G_q}(\infty )

the

G
q

{G_q}

-rationals, and

R

−

G
q

(
∞
)

\mathbb {R} - {G_q}(\infty )

the

G
q

{G_q}

-irrationals. The problem we treat here is the approximation of

G
q

{G_q}

-irrationals by

G
q

{G_q}

-rationals. Let

M
(
α
)

M(\alpha )

be the upper bound of numbers c for which

|

α
−
k

/

m

|

>
1

/

c

m
2

|\alpha - k/m| > 1/c{m^2}

for all

G
q

{G_q}

-irrationals and infinitely many

k

/

m
∈

G
q

(
∞
)

k/m \in {G_q}(\infty )

. Set

h
q
′

=

inf
α

M
(
α
)

h_q’= {\inf _\alpha }M(\alpha )

. We call

h
q
′

h_q’

the Hurwitz constant for

G
q

{G_q}

. It is known that

h
q
′

=
2

h_q’= 2

, q even;

h
q
′

=
2

(
1
+

(
1
−

λ
q

/

2

)
2

)

1

/

2

h_q’= 2{(1 + {(1 - {\lambda _q}/2)^2})^{1/2}}

, q odd. In this paper we prove this result by using

λ
q

{\lambda _q}

-continued fractions, as developed previously by D. Rosen. Write
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha minus StartFraction upper P Subscript n minus 1 Baseline Over upper Q Subscript n minus 1 Baseline EndFraction equals StartFraction left-parenthesis negative 1 right-parenthesis Superscript n minus 1 Baseline epsilon 1 epsilon 2 midline-horizontal-ellipsis epsilon Subscript n Baseline Over m Subscript n minus 1 Baseline left-parenthesis alpha right-parenthesis upper Q Subscript n minus 1 Superscript 2 Baseline EndFraction comma">

α
−

P

n
−
1

Q

n
−
1

=

(
−
1
)

n
−
1

ε
1

ε
2

⋯

ε
n

m

n
−
1

(
α
)

Q

n
−
1

2

,

\alpha - \frac {{{P_{n - 1}}}}{{{Q_{n - 1}}}} = \frac {{{{( - 1)}^{n - 1}}{\varepsilon _1}{\varepsilon _2} \cdots {\varepsilon _n}}}{{{m_{n - 1}}(\alpha )Q_{n - 1}^2}},

\]
where

ε
i

=
±
1

{\varepsilon _i} = \pm 1

and

P
i

/

Q
i

{P_i}/{Q_i}

are the convergents of the

λ
q

{\lambda _q}

-continued fraction for

α
\alpha

. Then

M
(
α
)
=

lim
¯

n

m
n

(
α
)

M(\alpha ) = {\overline {\lim } _n}{m_n}(\alpha )

. We call

m
n

(
α
)

{m_n}(\alpha )

the local Hurwitz constant. In the final section we prove some results on the local Hurwitz constant. For example (Theorem 4), it is shown that if q is odd and

ε

n
+
1

=

ε

n
+
2

=
+
1

{\varepsilon _{n + 1}} = {\varepsilon _{n + 2}} = + 1

, then

m
i

≥

(

λ
q
2

+
4

)

1

/

2

>

h
q
′

{m_i} \geq {(\lambda _q^2 + 4)^{1/2}} > h_q’

for at least one of

i
=
n
−
1
,
n
,
n
+
1

i = n - 1,n,n + 1

.