Gaussian positional disorder has been introduced into the structure of various 2D and 3Dcrystal lattices. As the degree of positional disorder is increased, the weights of thehighest-k peaks in the orientationally-averaged structure factor are decreased first, eventuallyleaving only a single peak having significant weight, that with the lowestk-value. The orientationally-averaged real-space pair-correlation function for lattices withsuch high levels of positional disorder exhibits a corresponding power-law-damped series ofoscillations, with a single period equal to the separation between the furthest-separated,lowest-k lattice planes. This last surviving peak in the structure factor is at ak-value nearly identical to that of the experimentally-observed first sharp diffractionpeak for the corresponding real amorphous phases of such lattices (e.g. Si,SiO2). These results also have a bearing on the unsolved Gauss circle problem in mathematics.