High-Dimensional Signal Compression: Lattice Point Bounds and Metric Entropy

We study worst-case signal compression under an ${\mathrm{\ell }}^2$ energy constraint, with coordinate-dependent quantization precisions. The compression problem is reduced to counting lattice points in a diagonal ellipsoid. Under balanced precision profiles, we obtain explicit, dimension-dependent upper bounds on the logarithmic codebook size. The analysis refines Landau’s classical lattice point estimates using uniform Bessel bounds due to Olenko and explicit Abel summation.