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ESTIA TERMINATED

Exponential sums, translation invariance, and applications

Total Cost €

0

EC-Contrib. €

0

Partnership

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 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.

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

Project "ESTIA" data sheet

The following table provides information about the project.

Coordinator
UNIVERSITY OF BRISTOL 

Organization address
address: BEACON HOUSE QUEENS ROAD
city: BRISTOL
postcode: BS8 1QU
website: www.bristol.ac.uk

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

 Partnership

Take a look of project's partnership.

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

Map

 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.

 Publications

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
2019 J. W. BOBER, D. FRETWELL, G. MARTIN, T. D. WOOLEY
SMOOTH VALUES OF POLYNOMIALS
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
VINOGRADOV SYSTEMS WITH A SLICE OFF
published pages: 797-817, ISSN: 0025-5793, DOI: 10.1112/S0025579317000134
Mathematika 63/03 2019-09-27

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