Incidence estimates for tubes with application to Fourier analysis

If T is a collection of distinct tubes of length N, radius 1 and P is a collection of disjoint unit balls, what is the largest size of their incidences I(P,T)={(p,T):|p¿T|¿1/10,p¿P,T¿T}? We will use some basic Fourier analysis to give upper bounds on the number of incidences. Then we discuss how to use these upper bounds to solve some problems in Fourier analysis. This includes joint work with Larry Guth, Noam Solomon, and Ciprian Demeter.