Optimal quantization for uniform distributions on Cantor-like sets

In this paper, the problem of optimal quantization is solved for uniform distributions on some higher dimensional, not necessarily self-similar N−adic Cantor-like sets. The optimal codebooks are determined and the optimal quantization error is calculated. The existence of the quantization dimension is characterized and it is shown that the quantization coefficient does not exist. The special case of self-similarity is also discussed. The conditions imposed are a separation property of the distribution and strict monotonicity of the first N quantization error differences. Criteria for these conditions are proved and as special examples modified versions of classical fractal distributions are discussed.