限制零的m进制数集之Hausdorff测度

对于任意整数表示mkz的分数部分.给出了数集Fm的Hausdorff测度是Fm的HausdorfF维数.

Hausdorff Measure for the Set of $m$-Adic Numbers without Neighboring Zeroes

For any integer $m\geq2$, let$F_m=\big\{x\in0,1): \{m^kx\}\geq \frac{1}{m^2},k\in N\big\}$,where $\{m^kx\}$ is the fractional part of $m^kx$. This paper gives the Hausdorff measure of $F_m$:\,$H^s(F_m)=(\frac{m^2-2}{m^2-1})^s$, where $s=\log_m\frac{m-1+\sqrt{(m-1)^2+4(m-1)}}{2}$ is the Hausdorff dimension of $F_m$.