|
Abstract:
|
We prove several theorems in the intersection of harmonic analysis ,
combinatorics , probability and number theory . In the second section we use combinatorial methods to construct various sets with pathological combinatorial
properties . In particular , we answer a question of P . Erdos and V . Sos regarding unions of Sidon sets . In the third section we use incidence bounds and bilinear methods to prove several new endpoint restriction estimates for the Paraboloid over finite fields . In the fourth and fifth sections we study a variational maximal operators associated to orthonormal systems . Here we use probabilistic techniques to construct well -behaved rearrangements and base
changes . In the sixth section we apply our variational estimates to a problem in sieve theory . In the seventh section , motivated by applications to sieve theory , we disprove a maximal inequality related to multiplicative characters . |