Arithmetic analogues in harmonic analysis: Results related to Waring's problem

Abstract

During the past score of years, there has been renewed interest in the area of discrete analogues in harmonic analysis. This began with an observation of M. Reisz in his work on the Hilbert transform in 1928 that was carried over in the work of A. Calderon and A. Zygmund on singular integrals in 1952. After a lull for many years, renewed interest began with the work of J. Bourgain on ergodic theorems along polynomial sequences, and G. Arkhipov and K. Oskolkov on the discrete Hilbert transform along polynomial sequences in the late 1980's, injecting methods and ideas from analytic number theory into the area.

This thesis establishes results on arithmetic maximal functions. In the continuous setting, these maximal functions go back to the 1976 work of E. Stein on the spherical maximal function. The arithmetic analogue, the discrete spherical maximal function is much newer, being initially investigated by A. Magyar in 1997. These results build on the work of Magyar and Magyar--Stein--Wainger on the discrete spherical maximal function to study arithmetic maximal functions on the hypersurfaces arising in Waring's problem. We conjecture the sharp results and combine the recent progress By T. Wooley on Waring's problem with the methods of Magyar and Magyar--Stein--Wainger to improve on previously known bounds. Subsequently, we apply these results to ergodic theory and incidence theory.