Optimal quantization for some triadic uniform Cantor distributions with exact bounds

Abstract

Let $\{S_j : 1\leq j\leq 3\}$ be a set of three contractive similarity mappings such that $S_j(x)=rx+\frac {j-1}{2}(1-r)$ for all $x\in \mathbb R$, and $1\leq j\leq 3$, where $0<r<\frac 1 3$. Let $P=\sum_{j=1}^3 \frac 13 P\circ S_j^{-1}$. Then, $P$ is a unique Borel probability measure on $\mathbb R$ such that $P$ has support the Cantor set generated by the similarity mappings $S_j$ for $1\leq j\leq 3$. Let $r_0=0.1622776602$, and $r_1=0.2317626315$ (which are ten digit rational approximations of two real numbers). In this paper, for $0<r\leq r_0$, we give a general formula to determine the optimal sets of $n$-means and the $n$th quantization errors for the triadic uniform Cantor distribution $P$ for all positive integers $n\geq 2$. Previously, Roychowdhury gave an exact formula to determine the optimal sets of $n$-means and the $n$th quantization errors for the standard triadic Cantor distribution, i.e., when $r=\frac 15$. In this paper, we further show that $r=r_0$ is the greatest lower bound, and $r=r_1$ is the least upper bound of the range of $r$-values to which Roychowdhury formula extends. In addition, we show that for $0<r\leq r_1$ the quantization coefficient does not exist though the quantization dimension exists.