This project is concerned with various problems in additive combinatorics and in particular with their applications to the Kakeya problem. The Kakeya problem asks, loosely speaking, how small a set in Euclidean space can be if it contains a unit line segment in every direction. More precisely, one would like lower bounds on the Hausdorff dimension, and it is conjectured that the lower bound should be the dimension of the Euclidean space. The conjecture has long been known true in the plane, but is much more difficult in higher dimensions and has inspired a great deal of work using techniques from fields ranging from harmonic analysis to commutative algebra to logic. In a paper with Laba and Tao, the author helped develop the connection of the Kakeya problem to sum-product theory. This is the theory of lower bounds on the sumset and product set of some finite subset of a field (or ring.) Sum product theory has been developing quickly over the past decade, and we now hope we know enough to come close to resolution in dimension 3, using some recent techniques of Bourgain, Konyagin, and others. In particular, we hope to show that the Assouad dimension of a Kakeya set in three dimensions is 3.
From the earliest part of our mathematical educations, we learn that addition and multiplication are two of the most important mathematical processes, and that they are widely applicable to a number of real world problems. Surprisingly, not everything about the way in which addition and multiplication are connected is completely understood. We have recently learned that in a certain sense, addition and multiplication do not go well together - if a set does not expand much under addition, it must expand under multiplication, and vice versa. This principle is expressed in a number of "sum product" theorems. Sum Product theory has led to a lot of excitement in the last few years. It has applications in the combinatorics and geometric measure theory, the parts of mathematics for which it was developed, as well as in theoretical computer science, where expanding sets are used to generate pseudo-randomness. This project deals both with the basic theory of sums and products as well as with their deep mathematical applications. Continued work in this area is certain to lead to further applications.
This research grant, as far as intellectual merit was concerned, was the most successful of my career. It led to the solution of two major problems in combinatorics, which were central in their respective field and which had been worked on serious by a number of major mathematicians in the past. The first of these was the Erdos distinct distance problem, for which I obtained a nearly optimal bound, jointly with Larry Guth. Roughly speaking, the problem asks how few distinct distances are possible between a set of a fixed number of points in the plane. Is it possible to find an arrangement in the plane for which surprisingly few of the distances are distinct. Is there a particularly good arrangement allowing a remarkable number of coincidences to occur. We rule this out. The second problem was to improve a bound of Meshulam on Cap sets. Here we are working in a high dimensional finite geometry over the field with three elements. The question is how large can a subset in this geometry be which contains no lines. In work with Michael Bateman, we improved what had been for a long time the best upper bound on this problem. The question is interesting to mathematicians, because lines in this geometry are a good model for arithmetic progressions of length three, In this way, our work relates to a theorem of Roth and toFields medal winning work by Gowers on bounds on sets having no arithmetic progressions and by Green and Tao on arithmetic progressions in the primes. As far as broader impact, the proposal was invaluable in the training of graduate students, in particular my graduate student Andrew Tapay. Also, it permitted to present findings at conferences such as Eurocomb '12 and a conference on additive combinatorics organized by Ben Green in Gregynog. These types of activities are invaluable for disseminating the results of mathematical research.