We present high-angular-resolution optical I-band imaging of 451 late K to mid-M nearby stars. These observations have been performed with Astralux and FastCam using the lucky imaging technique. We found 70 companions with separations between 0.1 and 3.5 arcsec, out of which 28 are new discoveries. We derive a total binary fraction of 20.3+ 6.9- 5.2 per cent. Follow-up observations have been carried out for 16 of these new binaries and all of them were confirmed as physically associated systems. Parallaxes are available for 70 per cent of the binary sample, providing reliable estimates of physical separations. We have measured separations and position angles as an initial stage for a systematic follow-up aimed to determine orbital parameters.