The proof neatly combines together several powerful and modern tools in a new way: In this post, I thought I would sketch some of the key ideas used in the proof, though I will not give the full argument here the paper itself is largely self-contained, well motivated, and of only moderate length.
That is an acceptable alternative because it allows us to apply the algebraic method. For instance, one can check that andalthough the precise computation of rapidly becomes more difficult after this. In a historical achievement, Katz and Guth were able to reach the exponent of 1, heretofore thought to be impossible.
The problem is to determine what the minimum possible set of distances is," Katz said. A typical such cheat: Combinatorial geometry, a field that has far-reaching applications in areas as diverse as drug development, robot motion planning and computer graphics, examines discrete properties like symmetry, folding, packing, decomposition and tiling associated with combinations of geometric objects.
We either get extremely efficient decomposition that provides the incidence theorems we like, or the alternative, that the procedure fails and most of the lines lie in the zero set of a polynomial of fairly low degree.
Guth and Katz were able to combine the algebraic method with a result from topology called the polynomial ham sandwich theorem, which was used to create a cell decomposition that yielded the desired results when most points were in the interiors of the cells, while the alternative case could be handled by the algebraic method.
A related such cheat: By considering the slightly more sophisticated example of a lattice grid assuming that is a square number for simplicityand using some analytic number theory, one can obtain the slightly better asymptotic bound. For more information or to speak with Katz, please contact Steve Chaplin, University Communications, at or stjchap indiana.
By considering points in arithmetic progression, we see that. A key new insight is that the polynomial method and more specifically, the polynomial Ham Sandwich theorem, also discussed previously on this blog can be used to efficiently create cells.
Building upon decades of work by others, Katz and Guth examined the problem within a group of rigid motions of the plane and used Euclidean geometry to view the distance problem as a three-dimensional linear one. Using that framework, Katz and Guth then implemented the polynomial ham sandwich theorem to create the new kind of cell decomposition that left points in the plane either in the interior of cells or on the walls of the cells.
On the other hand, lower bounds are more difficult to obtain. Until now that prize had gone unclaimed. Indeed, previously to last week, the best lower bound known was approximately Very recently, though, Guth and Katz have obtained a near-optimal result: In particular I will not go through all the various cases of configuration types that one has to deal with in the full argument, but only some illustrative special cases.
Theorem 2 One has. Introducing two new ideas to the existing proof, the pair exponentially improved upon the most recently described lower bound of.erdos_dist. Next: Bibliography. A Webpage on The Erdos Distance Problem. This one looks like it uses more advanced math. Katz and Tardos Elekes and Sharir ES.
There are no new lower bounds on in this paper; however, Guth and Katz use their framework to obtain their result. Guth and Katz. See GK. LOWER BOUND FOR and. The recent paper On the Erdos distinct distance problem in the plane Authors: Larry Guth, Nets Hawk Katz prodded me to get a non-trivial example.
Here is what I cannot find: an example of 12 erdos mint-body.comatorics. In [ES], Elekes and Sharir introduced a completely new approach to the distinct distance problem, which uses the symmetries of the problem in a novel way.
* Guth and Katz's recent work on the Erdos distinct distance problem, which beautifully synthesizes all of the above. Brown's construction. Application to combinatorial geometry: the unit distance problem in R^2. Lecture 4. Ramsey theory, part 1.
The party problem. Bounding the classical diagonal Ramsey numbers. Linear algebra. UNEXPECTED APPLICATIONS OF POLYNOMIALS IN COMBINATORICS LARRY GUTH In the last six years, several combinatorics problems have been solved in.
InKatz along with Larry Guth published the results of their collaborative effort to solve the Erdős distinct distances problem, in which they found a "near-optimal" result, proving that a set of points in the plane has at least / distinct distances.Download