Sets preserved by a large subgroup of the special linear group

Abstract Let be a subset of the affine plane over a finite field . We bound the size of the subgroup of that preserves . Specifically, there exists a constant such that for any , if is preserved by more than elements of , then is contained in a line through the origin. This result is sharp in general, and will be proved by using combinatorial arguments and applying a point‐line incidence bound in due to Mockenhaupt and Tao (2004).