## Final Report Summary - AAS (Approximate algebraic structure and applications)

The aim of the project was to advance the mathematical area of additive combinatorics and to explore its connection with other parts of mathematics. To create a unified project theme, the title of ``approximate algebraic structure'' was chosen. Many key advances in the last decade or so can be understood in terms of the desire to properly understand ``approximate'' algebraic objects. For example, the theory of Gowers norms and higher-order Fourier analysis may be understood in terms of approximate polynomials.

Over 20 papers were produced during the awarding period*, and there were several research highlights. Many of them did indeed come from a study of approximate algebraic structure as outlined in the project proposal 5 years ago. Others, happily, came about from other advances during the last 5 years, which could not have been foreseen.

*Around half of these are preprints. All are available on the arxiv and have been submitted for publication. A list of these publications can be provided on request.

Key research highlights sponsored by the grant include:

- The PI and Ford, Konyagin, Maynard and Tao proved that there are large gaps between primes, giving the first significant improvement on this basic question since 1938;

- The PI and Eberhard and Manners proved a 50-year old conjecture of Paul Erdos on sum-free sets (there exists a set of n integers with no sum-free subset bigger than about n/3);

- Bob Hough proved Erdos's covering congruences conjecture, showing that one cannot cover the integers by congruences a mod q, one for each sufficiently large q;

- The PI and Tao solved an old problem in combinatorial geometry to do with the number of ordinary lines (lines containing just two points), and proving a link to the theory of cubic curves;

- The PI, Eberhard and Ford solved two problems about random permutations using ideas form statistical number theory, for example obtaining for the first time the correct behaviour of p(k), the probability that a random permutation fixes some set of size k;

- The PI, Breuillard, Guralnick and Tao used the theory of approximate groups to give a detailed study of the expansion phenomenon in groups of Lie type;

- The PI and Sanders solved the "mod p" version of an old conjecture of Hindman by showing that colourings of Z/pZ contain monochromatic quadruples x, y, x+y, xy.

- The PI and Lindqvist completely clarified the Ramsey theory of the nonlinear equation x + y = z^2, proving that any 2-colouring of the integers contains a solution to this equation, but 3-colourings need not.

Over 20 papers were produced during the awarding period*, and there were several research highlights. Many of them did indeed come from a study of approximate algebraic structure as outlined in the project proposal 5 years ago. Others, happily, came about from other advances during the last 5 years, which could not have been foreseen.

*Around half of these are preprints. All are available on the arxiv and have been submitted for publication. A list of these publications can be provided on request.

Key research highlights sponsored by the grant include:

- The PI and Ford, Konyagin, Maynard and Tao proved that there are large gaps between primes, giving the first significant improvement on this basic question since 1938;

- The PI and Eberhard and Manners proved a 50-year old conjecture of Paul Erdos on sum-free sets (there exists a set of n integers with no sum-free subset bigger than about n/3);

- Bob Hough proved Erdos's covering congruences conjecture, showing that one cannot cover the integers by congruences a mod q, one for each sufficiently large q;

- The PI and Tao solved an old problem in combinatorial geometry to do with the number of ordinary lines (lines containing just two points), and proving a link to the theory of cubic curves;

- The PI, Eberhard and Ford solved two problems about random permutations using ideas form statistical number theory, for example obtaining for the first time the correct behaviour of p(k), the probability that a random permutation fixes some set of size k;

- The PI, Breuillard, Guralnick and Tao used the theory of approximate groups to give a detailed study of the expansion phenomenon in groups of Lie type;

- The PI and Sanders solved the "mod p" version of an old conjecture of Hindman by showing that colourings of Z/pZ contain monochromatic quadruples x, y, x+y, xy.

- The PI and Lindqvist completely clarified the Ramsey theory of the nonlinear equation x + y = z^2, proving that any 2-colouring of the integers contains a solution to this equation, but 3-colourings need not.