| Over the past decade, discrete geometry research has flourished with clever uses of algebraic methods. The polynomial method has had a deep impact on a wide collection of results in combinatorics, such as tight asymptotic lower bounds on finite field Kakeya and Nikodym sets, near optimal lower bound for Erdos' distinct distances problem, and improved bounds for cap sets. Spectral methods and rank bounds for matrices have shed new light on improved bounds for point-line incidences, subspace intersections and graph rigidity. This thesis is focused on developing new ways to improve on these techniques, and applying them in a few discrete geometry settings: 1. Techniques such as matrix scaling and rank bounds for design matrices have recently found beautiful applications for understanding configurations of points and lines over the complex numbers. In particular, they give a different proof of Kelly's Theorem, which says that any configuration of points in complex space must either be contained in a plane or have a line passing through exactly 2 of those points, called an ordinary line. We expand on these techniques to prove the first quantitative bounds for the number of ordinary lines in a non planar configuration of points in complex space. 2. In 2008, showed in a breakthrough result that a Kakeya set over finite fields has an asymptotically tight lower bound using the polynomial method. Later in 2008, Saraf and Sudan improved on the polynomial method by interpolating a polynomial that vanishes with high multiplicity on points of the Kakeya set. We further enhance the polynomial method by introducing the notion of ``fractional multiplicity," and use this improvement to obtain a better lower bound for finite field Kakeya sets in 3 dimensions. 3. While studying these 3-dimensional finite field Kakeya sets, we considered the related 3-dimensional finite field Nikodym sets. Previously, the lower bound for a 3-dimensional finite field Nikdoym sets was also obtained using the polynomial method, and had the same lower bound as for a Kakeya set. We achieve a better lower bound for 3-dimensional finite field Nikodym sets, thus separating it from the Kakeya set lower bound. |
