Exponential sums, translation invariance, and applications

Project Information

ESTIA

Grant agreement ID: 695223

DOI

10.3030/695223

Project terminated on 25 September 2019

EC signature date 20 April 2016

Start date 1 September 2016

End date 31 August 2021

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 873 483,00

EU contribution

€ 1 873 483,00

Coordinated by

UNIVERSITY OF BRISTOL
United Kingdom

Periodic Reporting for period 3 - ESTIA (Exponential sums, translation invariance, and applications)

Reporting period: 2019-09-01 to 2021-02-28

Summary of the context and overall objectives of the project

Exponential sums are fundamental throughout (analytic) number theory, and are key to the robustness of applications in theoretical computer science, cryptography, and so on. They are the primary tool for testing the equidistribution (apparent “randomness”) of number theoretic sequences. For a century, bounds for such sums of degree 3 or more have fallen far short of those conjectured to hold.

The landscape for exponential sums changed decisively in late 2010, when the proposer devised the “efficient congruencing” method. As a result, mean value estimates associated with translation invariant systems came within a whisker of the main conjectures. Very significant progress resulted in such Diophantine applications as Waring's problem (when can an integer be represented as a sum of kth powers of positive integers?), the validity of the Hasse principle for systems of diagonal equations (when can an integral linear combination of k-th powers of positive integers vanish?), and equidistribution of polynomial sequences mod 1 (under what circumstances does a polynomial with real coefficients appear random for random choices of integral input?).

The proposed research sought to establish and largely achieved the most optimistic versions of estimates conjectured to hold for moments of exponential sums in the widest possible setting, and in particular:
(i) to generalise efficient congruencing to approximately translation invariant systems involving polynomials not merely involving ascending powers, and explore consequent applications to Diophantine problems such as Waring's problem, restriction problems from discrete Fourier analysis, and bounds for the Riemann zeta function within the critical strip;
(ii) to derive robust and sharp estimates in arbitrary number fields, so that applications to Diophantine problems in number fields may be derived without problematic dependence on the degree of the number field over the rationals;
(iii) to extend the method to the multidimensional setting relevant to the investigation of local-global principles for spaces of rational morphisms from rational curves to diagonal hypersurfaces;
(iv) to explore the application of efficient congruencing over function fields where the ground field is a finite field, in particular as a vehicle for establishing estimates of use in randomness extractors;
(v) to investigate the potential use of higher degree translation invariance in generalising Gowers norms.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The PI's paper “Nested efficient congruencing and relatives of Vinogradov’s mean value theorem” (85pp, in press, Proceedings of the London Mathematical Society) fully resolves the main conjecture in Vinogradov's mean value theorem. In fact, given any system of k independent polynomials having integer coefficients, the 2s-moments of their weighted exponential sum exhibit square-root cancellation for s as large as k(k+1)/2. This result for the Vinogradov system t,t^2,...,t^k had been almost resolved in previous work of the author, and the small defect was resolved and main conjecture obtained in the work of Bourgain, Demeter and Guth (Annals, 2016). While the latter paper makes use of multilinear Kakeya inequalities restricted to discrete sets of real points, our work avoids such restrictions and extends to discrete sets of points in valued fields equipped with appropriate characters. We thus prove the main conjecture for Vinogradov's mean value theorem in number fields and function fields.

A key idea of nested efficient congruencing method is a weakened analogue of Hensel's lemma applicable to systems of equations in moments of exponential sums. This can be construed as a p-adic l^2-decoupling result, in the language of Bourgain et al. A precise analogue of this decoupling result has been achieved, and its consequences in higher dimensional problems is being explored. The idea also extends to systems of equations of only partial translation-dilation invariant character. Here, the p-adic interpretation fosters an approach which comes close to delivering the main conjecture. The ultimate objective is a "theory of everything" for mean values of exponential sums in arbitrarily many variables, and this should be amenable to analysis via efficient congruencing, delivering optimal estimates for the moments of associated exponential sums. Moreover, such moments would seem to exploit nothing more than linear structure inductively layered to create these higher degree systems.

The project finished on 11th August 2019 after slightly less than 3 years when the PI left Bristol (over Brexit issues) to move to Purdue University.

Mean Value Theorem