The project started on March 1st 2013 and will last until February 28, 2018.

Regularity and irregularity plays a central role in mathematics. In the present research proposal we will select problems from combinatorics
and number theory (including additive combinatorics), where regularity and irregularity appear. In some cases we have to deal, e.g., with
arbitrary finite or infinite subsets of natural numbers, where the only information we have is their cardinality, namely, that they are of
positive (lower asymptotic) density within the set of all natural numbers or within the interval [1,N] for a large N. In other cases we consider
an arbitrary distribution of n points within the unit square, where all we know is the density of our point set. The goal is often to show that
certain configurations appear within the arbitrary set of numbers or points. These configurations definitely appear in a random set of
numbers or points, but we have to show this for an arbitrary set of numbers or points with certain general properties. In order to reach our
goal one can use two well-known methods. The first one is deterministic, often some kind of greedy algorithm. The second is the
probabilistic method of Erds, which shows that almost all arrangements of the given points or numbers (or graphs) fulfill the wanted
property. A third method, the so called pseudorandom method, was initiated by the PI (together with M. Ajtai and J. Komlós), uses a
combination of these. In other cases we have a deterministic set of numbers with certain quasi-random properties, for example, the primes.
Randomness was the key idea in the recent breakthrough of Green and Tao, in proving that primes contain arbitrarily long arithmetic
progressions.

Our main fields of research are the following:

- finite or infinite sequences of integers;
- difference sets and Fourier analysis;
- graph and hypergraph embedding theorems;
- distribution of points in the plane and in the unit square;
- regularities and irregularities in the distribution of primes.

If you are interested in , please contact szem.endre@renyi.mta.hu

## Staff:

- Principal Investigator: Prof. Endre Szemeredi
- Prof. János Pintz
- Prof. Imre Ruzsa
- András Bíró
- Dr. Máté Matolcsi
- Dr. Miklós Simonovits
- Dr. József Solymosi
- Dr. Gergely Harcos
- Béla Csaba

## Visiting researchers:

- Oliver Roche Newton
- Brian Michael Cook