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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.

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

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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