This paper studies circular chromatic numbers and fractional chromatic numbers of distance graphs G(Z , D) for various distance sets D. In particular, we determine these numbers for those D sets of size two, for some special D sets of size three, for D = {1, 2, . . . , m, n} with 1 ≤ m < n, for D = {q, q + 1, . . . , p} with q ≤ p, and for D = {1, 2, . . . , m} − {k} with 1 ≤ k ≤ m.
Let
[
a
1
(
x
)
,
a
2
(
x
)
,
…
]
[a_1(x),a_2(x),\ldots ]
be the continued fraction expansion of
x
∈
[
0
,
1
)
x\in [0,1)
and let
q
n
(
x
)
q_n(x)
be the denominator of the
n
n
th convergent. Recently, Hussain-Kleinbock-Wadleigh-Wang (2018) showed that for any
τ
≥
0
,
\tau \ge 0,
the set
D
c
(
τ
)
=
{
x
∈
[
0
,
1
)
:
lim sup
n
→
∞
log
(
a
n
(
x
)
a
n
+
1
(
x
)
)
log
q
n
(
x
)
≥
τ
}
\begin{equation*} D^{c}(\tau )=\Big \{x\in [0,1): \limsup \limits _{n\rightarrow \infty }\frac {\log \big (a_n(x)a_{n+1}(x)\big )}{\log q_n(x)}\ge \tau \Big \} \end{equation*}
is of Hausdorff dimension
2
τ
+
2
.
\frac {2}{\tau +2}.
In this note, we study the Hausdorff dimension of the set
a
m
p
;
F
(
τ
)
=
{
x
∈
[
0
,
1
)
:
lim
n
→
∞
log
(
a
n
(
x
)
a
n
+
1
(
x
)
)
log
q
n
(
x
)
=
τ
}
.
\begin{align*} &F(\tau )=\Big \{x\in [0,1): \lim \limits _{n\rightarrow \infty }\frac {\log \big (a_n(x)a_{n+1}(x)\big )}{\log q_n(x)}=\tau \Big \}. \end{align*}
It is proved that the set
F
(
τ
)
F(\tau )
has Hausdorff dimension
1
1
or
2
τ
+
τ
2
+
4
+
2
\frac {2}{\tau +\sqrt {\tau ^2+4}+2}
according as
τ
=
0
\tau =0
or
τ
>
0.
\tau >0.
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