AI 中文总结
本文研究慢斐波那契游走的间隙谱与密度，证明Chung等人关于$D_\ell=U_\ell$的猜想对$\ell=3$成立、对$\ell=4$不成立，确定了三阶和四阶间隙谱，明确了相关集合的对数密度性质及$\ell=1$时的精确对数密度。
AI 中文摘要
设$F_1=F_2=1$，且对$t\geq1$有$F_{t+2}=F_{t+1}+F_t$。对每个$n\geq2$，存在唯一整数$a,b,t$使得$n=aF_t+bF_{t-1}$，其中$t\geq2$且$1\leq a\leq b\leq F_t$。初始对为$(b,a)$的斐波那契游走会尽可能晚到达$n$，且该游走中$n$之后的项，当$t$为偶数时为$\lfloor\phi n\rfloor$，当$t$为奇数时为$\lceil\phi n\rceil$，其中$\phi=(1+\sqrt{5})/2$。设$D=\{d_1<d_2<\cdots\}$和$U=\{u_1<u_2<\cdots\}$分别对应$t$为偶数和奇数的集合。对$\ell,m\geq1$，定义$D_\ell=\{d_{k+\ell}-d_k:k\geq1\}$，$U_\ell=\{u_{k+\ell}-u_k:k\geq1\}$，$D_\ell(m)=\{d_k:d_{k+\ell}-d_k=m\}$，$U_\ell(m)=\{u_k:u_{k+\ell}-u_k=m\}$。Chung、Graham和Siro猜想对所有$\ell$有$D_\ell=U_\ell$，并询问$D_\ell(m)$和$U_\ell(m)$的密度，尤其当$\ell=1$时的情况。本文确定了三阶和四阶间隙谱，证明该猜想对$\ell=3$成立但对$\ell=4$不成立；还通过刻画$D_\ell(m)$和$U_\ell(m)$何时具有自然密度，回答了他们的密度问题，并证明其对数密度始终存在且相等，对$\ell=1$还给出了精确的对数密度。
英文摘要
Let $F_1=F_2=1$ and $F_{t+2}=F_{t+1}+F_t$ for $t\geq1$. For every $n\geq2$, there are unique integers $a,b,t$ such that $n=aF_t+bF_{t-1}$ with $t\geq2$ and $1\leq a\leq b\leq F_t$. The Fibonacci walk with initial pair $(b,a)$ reaches $n$ as late as possible, and the term following $n$ in this walk is $\lfloorϕn\rfloor$ when $t$ is even and $\lceilϕn\rceil$ when $t$ is odd, where $ϕ=(1+\sqrt5)/2$. Let $D=\{d_1<d_2<\cdots\}$ and $U=\{u_1<u_2<\cdots\}$ be the sets corresponding to even and odd $t$, respectively. For $\ell,m\geq1$, define $D_\ell=\{d_{k+\ell}-d_k:k\geq1\}$, $U_\ell=\{u_{k+\ell}-u_k:k\geq1\}$, $D_\ell(m)=\{d_k:d_{k+\ell}-d_k=m\}$ and $U_\ell(m)=\{u_k:u_{k+\ell}-u_k=m\}$. Chung, Graham and Spiro conjectured that $D_\ell=U_\ell$ for all $\ell$, and asked for the densities of $D_\ell(m)$ and $U_\ell(m)$, especially when $\ell=1$. In this paper, we determine the third and fourth order gap spectra, and show that the conjecture holds for $\ell=3$ but fails for $\ell=4$. We also answer their density question by characterizing when $D_\ell(m)$ and $U_\ell(m)$ have natural densities and proving that their logarithmic densities always exist and are equal. For $\ell=1$, we give the exact logarithmic densities.
Comments12 pages
