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Exponential sums, translation invariance, and applications

Total Cost €


EC-Contrib. €






 ESTIA project word cloud

Explore the words cloud of the ESTIA project. It provides you a very rough idea of what is the project "ESTIA" about.

fallen    generalising    title    translation    gowers    finite    spaces    theory    computer    devised    approximately    changed    vehicle    ed    equations    rational    exponential    landscape    norms    extend    progress    science    fundamental    congruencing    critical    invariant    randomness    mean    offers    hitherto    robustness    sums    investigation    theoretical    equidistribution    diophantine    strip    principles    community    fourier    decisively    inaccessible    little    tool    waring    generalise    establishing    estimating    analytic    global    extractors    discrete    function    degree    curves    hypersurfaces    local    morphisms    apparent    setting    invariance    century    fundamentally    mod    whisker    efficient    2010    coefficients    ground    cryptography    primary    restriction    explore    generality    multidimensional    theoretic    hasse    zeta    conjectured    hold    polynomial    conjectures    validity    bounds    diagonal    moments    riemann    sequences    proposer    estimates    summary    consequent   

Project "ESTIA" data sheet

The following table provides information about the project.


Organization address
postcode: BS8 1QU

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country United Kingdom [UK]
 Total cost 1˙873˙483 €
 EC max contribution 1˙873˙483 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2015-AdG
 Funding Scheme ERC-ADG
 Starting year 2016
 Duration (year-month-day) from 2016-09-01   to  2021-08-31


Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 
1    UNIVERSITY OF BRISTOL UK (BRISTOL) coordinator 1˙873˙483.00


 Project objective

Title: Exponential Sums, Translation Invariance, and Applications.

Short Summary: 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 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 are now within a whisker of the main conjectures. Very significant progress has resulted in such Diophantine applications as Waring's problem, the validity of the Hasse principle for systems of diagonal equations, and equidistribution of polynomial sequences mod 1.

It is little understood in the wider community that efficient congruencing offers a fundamentally new approach to estimating moments of Fourier coefficients of wide generality, with hitherto inaccessible applications. We propose:

(i) to generalise efficient congruencing to approximately translation invariant systems, 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 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;

(iii) 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;

(iv) to investigate the potential use of higher degree translation invariance in generalising Gowers norms.


year authors and title journal last update
List of publications.
2020 Akshat Mudgal
Difference sets in higher dimensions
published pages: , ISSN: 9999-9994, DOI:
Preprint (submitted) 2020-03-24
2020 Javier Pliego Garcia
A uniform bound in Waring\'s problem over some diagonal forms
published pages: , ISSN: 9999-9994, DOI:
Preprint 2020-03-24
2020 Akshat Mudgal
Sum-product estimates for diagonal matrices
published pages: , ISSN: 9999-9994, DOI:
Preprint (submitted) 2020-03-24
2020 Javier Pliego Garcia
On squares of sums of three cubes
published pages: , ISSN: 9999-9994, DOI:
Quarterly journal of Mathematics (submitted February 2020) 2020-03-24
2020 J. Brandes, S. T. Parsell, C. Poulias, G. Shakan and R. C. Vaughan
On generating functions in additive number theory, II: Lower-order terms and applications to PDEs
published pages: , ISSN: 9999-9994, DOI:
Submitted January 2020 2020-03-24
2020 K. Hughes and T. D. Wooley
Discrete restriction for (x,x^3) and related topics
published pages: , ISSN: 9999-9994, DOI:
Duke Math. Journal (submitted December 2019) 2020-03-24
2019 Akshat Mudgal
Sums of Linear Transformations in Higher Dimensions
published pages: 965-984, ISSN: 0033-5606, DOI: 10.1093/qmath/haz006
The Quarterly Journal of Mathematics 70/3 2020-03-24
2020 Javier Pliego Garcia
On Waring\'s problem in sums of three cubes
published pages: , ISSN: 9999-9994, DOI:
International Mathematics Research Notices (submitted February 2020) 2020-03-24
2020 Javier Pliego Garcia
On Waring\'s problem in sums of three cubes of smaller powers
published pages: , ISSN: 9999-9994, DOI:
Journal of the London Mathematical Society (submitted January 2020) 2020-03-24
2019 Jörg Brüdern, Trevor D. Wooley
An instance where the major and minor arc integrals meet
published pages: 1113-1128, ISSN: 0024-6093, DOI: 10.1112/blms.12291
Bulletin of the London Mathematical Society 51/6 2020-03-24
2020 Akshat Mudgal
Arithmetic combinatorics on Vinogradov systems
published pages: , ISSN: 0002-9947, DOI:
Transactions of the American Mathematical Society (accepted, to appear) 2020-03-24
2020 Brandes, Julia; Wooley, Trevor
Optimal mean value estimates beyond Vinogradov\'s mean value theorem
published pages: , ISSN: 9999-9994, DOI:
International Mathematics Research Notices (submitted January 2019) 1 2020-03-24
2020 C. Poulias
Diagonal Diophantine inequalities of fractional degree
published pages: , ISSN: 9999-9994, DOI:
Preprint 2020-03-24
2018 Trevor D. Wooley
Nested efficient congruencing and relatives of Vinogradov\'s mean value theorem
published pages: 942-1016, ISSN: 0024-6115, DOI: 10.1112/plms.12204
Proceedings of the London Mathematical Society 118/4 2019-09-27
published pages: 1-17, ISSN: 1446-7887, DOI: 10.1017/S1446788718000320
Journal of the Australian Mathematical Society 2019-09-27
2017 Julia Brandes, Trevor D. Wooley
published pages: 797-817, ISSN: 0025-5793, DOI: 10.1112/S0025579317000134
Mathematika 63/03 2019-09-27

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