Analytic and Additive Number Theory


Math 8440




Textbook:      There is no textbook for the course, but here are some recommended books. I will post notes and links to the various topics.

Description:  The aim of the course is to give an introduction to the Hardy-Littlewood circle method of exponential sums, and also the more recent Goldston-Yildirim-Pintz sieve.
                       We will discuss various classical and more recent applications.  Topics may include but may not be limited to




I. The circle method

We introduce the circle method and apply it to calculate the asymptotic number and distribution of integer points on spheres.

           Notes                                                                                                       Supplementary Notes and Papers                                                                                                    Problems and Projects
            
Sums of squares 1                                                                                    Weyl sums & Sarkozy's Theorem
Sums of squares 2                                                                                              
Fourier Transform of integer points on spheres                                      Uniformity of distribution of lattice points on spheres
Equi-distribution of integer points on spheres
     




                                                                                                                                             II. Distance sets of large sets in R^d and Z^d.

            We apply the structural information about the Fourier transform of integer points on spheres prove a discrete analogue of the Katznelson-Weiss theorem on the distance sets of large sets in Euclidean spaces.
            This may be viewed as a result in geometric Ramsey theory, where one studies geometric configuration is sets of positive upper density.



           Notes                                                                                                    Supplementary Notes and Papers                                                                                                      Problems and Projects

         
Distance sets 1                                                                                      A Szemerédi type theorem for sets of positive density in R^k
                                                                                                                        
         
Distance sets 2                                                                                      On distance sets of large sets of integer points
                                                                                                                        k-point configurations in sets of positive density of Z^n
          Distance sets 3




                                                                                                                                                         III. Diophantine equations  


          Notes                                                                                                     Supplementary Notes and Papers                                                               

         Diophantine equations I                                                                         Davenport: Geometry of numbers

         Diophantine equations II

         Diophantine equations III



  
                                                                                                                                                            IVSieve Methods


         Notes                                                                                                    Supplementary Notes and Papers

                                                                                                                     
Green-Tao: Correlation estimates for sieve weights