Error Bounds for High-resolution Quantization with Rényi-α-entropy Constraints

We consider the problem of optimal quantization with norm exponent r > 0 for Borel probability measures on R under constrained Rényi-αentropy of the quantizers. If the bound on the entropy becomes large, then sharp asymptotics for the optimal quantization error are well-known in the special cases α = 0 (memory-constrained quantization) and α = 1 (Shannon-entropy-constrained quantization). In this paper we determine sharp asymptotics for the optimal quantization error under large entropy bound with entropy parameter α ∈ [1 + r/d,∞]. For α ∈ [0, 1 + r/d[ we specify the asymptotical order of the optimal quantization error under large entropy bound. The optimal quantization error is decreasing exponentially fast with the entropy bound and the exact rate is determined for all α ∈ [0,∞].