Grate the beets and carrots in a food processor using a medium-coarse grating blade.
Cut the cabbage into chunks small enough to fit through your processor’s feed tube and grate with the same blade.
Combine the grated beets and cabbage in a serving container and dress to taste with the lemon juice and syrup. Stir well, then add the dill and salt and stir again.
Let the salad stand for at least 15 minutes or so before serving. Taste again and adjust the lemon and sweetener if needed, then serve.