High-resolution scalar quantization with R\'enyi entropy constraint

We consider optimal scalar quantization with $r$th power distortion and constrained R'enyi entropy of order $\alpha$. For sources with an absolutely continuous distribution the high rate asymptotics of the quantizer distortion has long been known for $\alpha =0$ (fixed-rate quantization) and $\text{al}pha=1$ (entropy-constrained quantization). For a large class of absolutely continuous source distributions we determine the sharp asymptotics of the optimal quantization distortion for $\alpha \in [-\mathrm{\infty },0)\cup (0,1)$. The achievability proof is based on finding (asymptotically) optimal quantizers via the companding approach, and is thus constructive.