Scalar quantization with Rényi entropy constraint

We consider optimal scalar quantization with rth power distortion and constrained Rényi entropy of order α. For sources with absolutely continuous distributions the high rate asymptotics of the quantizer distortion has long been known for α = 0 (fixed-rate quantization) and α = 1 (entropy-constrained quantization). For a large class of absolutely continuous source distributions we determine the sharp asymptotics of the optimal quantization distortion for Rényi entropy constraints of order α ∈ [-∈, 0) ∪ (0; 1). The proof of achievability is based on companding quantization and is thus constructive.