论文标题
通过斐波那契序列着色的平衡序列的渐近重复阈值的上限
An upper bound on asymptotic repetitive threshold of balanced sequences via colouring of the Fibonacci sequence
论文作者
论文摘要
我们通过合适的恒定间隙序列为斐波那契序列着色，以在$ d $ ary平衡序列的渐近重复阈值上提供上限。 $ d = 2、4 $和$ 8 $获得了界限，我们猜想它发生在无限的许多$ d $中。 我们的界限揭示了重复阈值的行为和平衡序列的渐近重复阈值的本质差异。 $ d $ - ary平衡序列的重复阈值至少为$ 1+\ frac {1} {D-2} $，对于每个$ d \ geq 3 $。相比之下，我们的界限意味着$ d $ - ary平衡序列的渐近重复阈值最多为$ 1+\ frac {τ^3} {2^{d-3}} $，对于每个$ d \ geq 2 $，其中$τ$的$ d \ geq 2 $是金色的意思。
We colour the Fibonacci sequence by suitable constant gap sequences to provide an upper bound on the asymptotic repetitive threshold of $d$-ary balanced sequences. The bound is attained for $d=2, 4$ and $8$ and we conjecture that it happens for infinitely many even $d$'s. Our bound reveals an essential difference in behavior of the repetitive threshold and the asymptotic repetitive threshold of balanced sequences. The repetitive threshold of $d$-ary balanced sequences is known to be at least $1+\frac{1}{d-2}$ for each $d \geq 3$. In contrast, our bound implies that the asymptotic repetitive threshold of $d$-ary balanced sequences is at most $1+\frac{τ^3}{2^{d-3}}$ for each $d\geq 2$, where $τ$ is the golden mean.
