Regularity and Irregularity in Combinatorics and Number Theory

Obiettivo

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 Erdős, 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. We will deal with 6 groups of problems: (i) finite or infinite sequences of integers, (ii) difference sets and Fourier analysis, (iii) graph and hypergraph embedding theorems, (iv) Ramsey theory, (v) distribution of points in the plane and in the unit square, (vi) regularities and irregularities in the distribution of primes.

Campo scientifico

Meccanismo di finanziamento

ERC-AG - ERC Advanced Grant

Istituzione ospitante

HUN-REN RENYI ALFRED MATEMATIKAI KUTATOINTEZET

Contributo UE

€ 1 776 000,00

Indirizzo

REALTANODA STREET 13-15
1053 Budapest
Ungheria

Regione

Közép-Magyarország Budapest Budapest

Tipo di attività

Other

Contatto amministrativo

Tiziana Del Viscio (Ms.)

Ricercatore principale

Endre Szemeredi (Prof.)

Collegamenti

Contatta l’organizzazione Sito web

Costo totale

Nessun dato

Beneficiari (1)