Sylvester-Gallai Type Theorems for Approximate Collinearity

Abstract

We study questions in incidence geometry where the precise position of points is ‘blurry’ (for example due to noise, inaccuracy or error). Thus lines are replaced by narrow tubes, and more generally affine subspaces are replaced by their small neighborhood. We show that the presence of a sufficiently large number of approximately collinear triples in a set of points in ℂ𝑑 implies that the points are close to a low dimensional affine subspace. This can be viewed as a stable variant of the Sylvester–Gallai theorem and its extensions. Building on the recently found connection between Sylvester–Gallai type theorems and complex locally correctable codes (LCCs), we define the new notion of stable LCCs, in which the (local) correction procedure can also handle small perturbations in the Euclidean metric. We prove that such stable codes with constant query complexity do not exist. No impossibility results were known in any such local setting for more than two queries.