Analytic Number Theory Motivated by Approximate Translation Invariance

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Exponential sums are Fourier series encoding arithmetic information. Pointwise bounds and mean values of such sums play a fundamental role throughout analytic number theory, and contribute the primary tool for testing equidistribution (apparent ``randomness'') of sequences underpinning many applications of number theory in theoretical computer science, cryptography, and so on. Until the last decade, despite almost a century of intense effort starting with the introduction by Hardy and Littlewood of their famous circle method, the main conjectures concerning mean values of exponential sums over polynomials remained unsolved in all but the very simplest cases involving linear and quadratic polynomials. A decade of dramatic progress has culminated in the last five years with the proof of the most ambitious conjectures concerning a central example of such mean value conjectures, that associated with Vinogradov's mean value theorem, on the one hand by Bourgain, Demeter and Guth via decoupling, and on the other by the proposer by means of nested efficient congruencing. In this project, the principal investigator will enhance, extend and exploit these very recent methods so as to obtain similarly decisive progress in an array of mean value conjectures having applications in quantitative arithmetic geometry and the wider theory of the Hardy-Littlewood method. This will contribute to the resolution of the main conjectures for translation-dilation invariant systems in many variables in full generality, including analogues of such conjectures involving mean values averaged over sets of small measure, beyond the reach of current technology. A graduate student will be trained in this important emerging area, and the proposer will work on a new text intended to provide an introduction to efficient congruencing for translation-dilation invariant systems as a vehicle for introducing modern developments in the circle method over the rational integers, number fields and function fields.Very recent advances in the understanding of mean values of exponential sums have delivered the Main Conjecture for Vinogradov's mean value. By orthogonality, this mean value is associated with a translation-dilation invariant Diophantine system. Despite this success, neither the decoupling method nor the nested efficient congruencing method currently address any but embryonic multivariable translation-dilation invariant systems. Moreover, they do not address corresponding mean values supported on subsets of the unit hypercube, and thus fail to provide useful estimates for either minor arcs or wide sets of major arcs of use in the Hardy-Littlewood method. This project will make decisive progress on this comprehensive theory, delivering the main conjectures concerning mean values of exponential sums associated not only with general translation-dilation invariant Diophantine systems, but also systems possessing only partial or approximate translation-dilation invariant structure. This will all be done in the quite general setting of number fields and function fields by adapting the proposer’s nested efficient congruencing methods. This flexible set of methods permits congruence information to be passed from one set of variables to another in multi-homogeneous settings, and this may be achieved even when working on restricted domains of integration. Amongst applications of these new estimates, the principal investigator will establish local-global principles for the existence of rational curves with rational coefficients on hypersurfaces possessing some measure of diagonal structure via the Hardy-Littlewood method.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.