Local gyrokinetic simulations solve the gyrokinetic equations with homogeneous background gradients, typically using a doubly periodic domain in the (x, y) plane (i.e. perpendicular to the field line). Spatial Fourier representations are almost universal in local gyrokinetic codes, and the wavevector-remap method was introduced in (Hammett et al, Bull Am Phys Soc VP1 136, (2006)) as a simple method for expressing the local gyrokinetic equations with a background shear flow in a Fourier representation. Although extensively applied, the wavevector-remap method has not been formally shown to converge, and suffers from known unphysicality when the solutions are plotted in real space (Fox et al PPCF 59, 044008). In this work, we use an analytic solution in slab geometry to demonstrate that wavevector-remap leads to incorrect smeared non-linear coupling between modes. We derive a correct, relatively simple method for solving local gyrokinetics in Fourier space with a background shear flow, and compare this to the wavevector-remap method. This allows us to show that the error in wavevector-remap can be seen as an incorrect rounding in wavenumber space in the nonlinear term. By making minor modifications to the nonlinear term, we implement the corrected wavevector-remap scheme in the GENE (T Dannert and F Jenko (2005), Physics of Plasmas 12, 072309) code and compare results of the original and corrected wavevector-remap for standard nonlinear benchmark cases. Certain physical phenomena are impacted by the errors in the original remap scheme, and these numerical artefacts do not reduce as system size increases: that is, original wavevector-remap scheme does not converge to the correct result.