Project work developed and explored the nested variant of the PI's multigrade efficient congruencing method. This p-adic concentration argument establishes conjectured estimates for the magnitude of the moments of the exponential sum over k-th degree polynomials. In contrast to the recent approach of Bourgain, Demeter and Guth via multilinear Kakeya estimates, the method applies to exponential sums over discrete sets of points in any valued field admitting appropriate characters. Thus, for example, it applies also to algebraic number fields and to function fields without any significant alteration.
Work in the first reporting period derived decoupling estimates in Waring's problem, and in character sum mean values. In addition, work joint with Yu-Ru Liu refined the function field estimates that stem from the simplest version of nested efficient congruencing for small characteristic. Significant work was also expended on approximate translation-dilation invariant systems, much joint with Julia Brandes. This supplies mean value estimates for systems in which there are missing degrees, and is part of the broader theory. On-going work promises to push these estimates within a whisker of the main conjecture. A major theme from the second reporting period concerns mean values for multidimensional systems having partial translation-dilation invariant structure only. This work, joint with Julia Brandes, has applications to earlier work on exponential sums in one variable in which there are missing degrees. Additional work joint with Kevin Hughes works toward providing explicit bounds for the epsilon-dependence in the new estimates for Vinogradov's mean value theorem. In a second direction, we provide p-adic analogues of precisely the l^2-decoupling bounds of Bourgain, Demeter and Guth, extending this to novel l^r-decoupling bounds with 1
Initial work of the students Konstantinos Poulias and Javier Pliego also concerns applications, to problems involving real fractional exponents, and problems in which the variables are sums of three cubes. Meanwhile, Akshat Mudgal has derived applications of interest in additive combinatorics, including an analogue of the Balog-Szemeredi-Gowers theorem for Vinogradov systems.
In summary, the project (after slightly less than 3 of its 5 intended years) has achieved a broad understanding of the efficient congruencing approach as applied in the setting of p-adic fields, number fields and function fields, and has extended somewhat successfully to systems not of translation-dilation invariant type. This work has been disseminated through conference and seminar talks, published journal papers, and uploads to the arXiv. Work still in preparation will be disseminated with full credit as appropriate.
