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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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In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph µ(G), we now call the Mycielskian of G, which has the same clique number as G and whose chromatic number equals χ(G)+1. Chang, Huang, and Zhu [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear] have investigated circular chromatic numbers of Mycielskians for several classes of graphs. In this article, we study circular chromatic numbers of Mycielskians for another class of graphs G d k . The main result is that χ c (µ(G d k )) = χ(µ(G d k )), which settles a problem raised in [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear, and X. Zhu, to appear]. As χ c (G d k ) = k d and χ(G d k ) = k d , consequently, there exist graphs G such that χ c (G) is as close to χ(G) − 1 as you want, but χ c (µ(G)) = χ(µ(G)).

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This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set: $$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$ where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$ ) shown by Hussain, Kleinbock, Wadleigh and Wang.