Imre Z. Ruzsa (born 23 July 1953) is a Hungarian

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

specializing in

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

Life

Ruzsa participated in the

International Mathematical Olympiad The International Mathematical Olympiad (IMO) is a mathematical olympiad for pre-university students, and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, except i ...

for Hungary, winning a silver medal in 1969, and two consecutive gold medals with perfect scores in 1970 and 1971. He graduated from the

Eötvös Loránd University Eötvös Loránd University ( hu, Eötvös Loránd Tudományegyetem, ELTE) is a Hungarian public research university based in Budapest. Founded in 1635, ELTE is one of the largest and most prestigious public higher education institutions in Hung ...

in 1976. Since then he has been at the

Alfréd Rényi Institute of Mathematics The Alfréd Rényi Institute of Mathematics ( hu, Rényi Alfréd Matematikai Kutatóintézet) is the research institute in mathematics of the Hungarian Academy of Sciences. It was created in 1950 by Alfréd Rényi, who directed it until his death. ...

of the

Hungarian Academy of Sciences The Hungarian Academy of Sciences ( hu, Magyar Tudományos Akadémia, MTA) is the most important and prestigious learned society of Hungary. Its seat is at the bank of the Danube in Budapest, between Széchenyi rakpart and Akadémia utca. Its ma ...

. He was awarded the

Rollo Davidson Prize The Rollo Davidson Prize is a prize awarded annually to early-career probabilists by the Rollo Davidson trustees. It is named after English mathematician Rollo Davidson (1944–1970). Rollo Davidson Trust In 1970, Rollo Davidson, a Fellow-elec ...

in 1988. He was elected corresponding member (1998) and member (2004) of the

. He was invited speaker at the

European Congress of Mathematics The European Congress of Mathematics (ECM) is the second largest international conference of the mathematics community, after the International Congresses of Mathematicians (ICM). The ECM are held every four years and are timed precisely betwee ...

at Stockholm, 2004, and in the Combinatorics section of the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...

in Madrid, 2006. In 2012 he became a fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

.List of Fellows of the American Mathematical Society
retrieved 2013-07-07.

Work

With

Endre Szemerédi Endre Szemerédi (; born August 21, 1940) is a Hungarian-American mathematician and computer scientist, working in the field of combinatorics and theoretical computer science. He has been the State of New Jersey Professor of computer science a ...

he proved subquadratic upper and lower bounds for the

Ruzsa–Szemerédi problem In combinatorial mathematics and extremal graph theory, the Ruzsa–Szemerédi problem or (6,3)-problem asks for the maximum number of edges in a graph in which every edge belongs to a unique triangle. Equivalently it asks for the maximum number ...

on the number of triples of points in which the union of any three triples contains at least seven points. He proved that an essential component has at least (log ''x'')^1+ε elements up to ''x'', for some ε > 0. On the other hand, for every ε > 0 there is an essential component that has at most (log ''x'')^1+ε elements up to ''x'', for every ''x''. He gave a new proof to

Freiman's theorem In additive combinatorics, Freiman's theorem is a central result which indicates the approximate structure of sets whose sumset is small. It roughly states that if , A+A, /, A, is small, then A can be contained in a small generalized arithmetic pr ...

. Ruzsa also showed the existence of a

Sidon sequence In number theory, a Sidon sequence is a sequence A=\ of natural numbers in which all pairwise sums a_i+a_j (for i\le j) are different. Sidon sequences are also called Sidon sets; they are named after the Hungarian mathematician Simon Sidon, who int ...

which has at least ''x''^0.41 elements up to ''x''. In a result complementing the

Erdős–Fuchs theorem In mathematics, in the area of additive number theory, the Erdős–Fuchs theorem is a statement about the number of ways that numbers can be represented as a sum of elements of a given additive basis, stating that the average order of this numbe ...

he showed that there exists a sequence ''a''₀, ''a''₁, ... of natural numbers such that for every ''n'' the number of solutions of the inequality ''a''_''i'' + ''a''_''j'' ≤ ''n'' is ''cn'' + ''O''(''n''^1/4log ''n'') for some ''c'' > 0.

Selected publications

* * * *

References

External links

Some of Ruzsa's papers
at the Rényi Institute * {{DEFAULTSORT:Ruzsa, Imre Z. 1953 births Living people Number theorists 20th-century Hungarian mathematicians 21st-century Hungarian mathematicians Members of the Hungarian Academy of Sciences Fellows of the American Mathematical Society International Mathematical Olympiad participants

Life

Work

Selected publications

See also

References

External links