Imre Z. Ruzsa
Encyclopedia
Imre Z. Ruzsa is a Hungarian mathematician
specializing in number theory
.
Ruzsa participated in the International Mathematical Olympiad
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 in 1976. Since then he has been at the Alfréd Rényi Institute of Mathematics
of the Hungarian Academy of Sciences
. He was awarded the Rollo Davidson Prize
in 1988. He was elected corresponding member (1998) and member (2004) of the Hungarian Academy of Sciences
. He was invited speaker at the European Congress of Mathematics
at Stockholm, 2004, and in the Combinatorics section of the International Congress of Mathematicians
in Madrid, 2006.
With Endre Szemerédi
he proved that on n points only o(n2) triples can be given such that the union of any 3 of them contains at least 7 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
. Ruzsa also showed the existence of a Sidon sequence which has at least x0.41 elements up to x.
In a result complementing the Erdős–Fuchs theorem
he showed that there exists a sequence a0, a1, ... of natural numbers such that for every n the number of solutions of the inequality ai + aj ≤ n is cn + O(n1/4log n) for some c > 0.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
specializing in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
.
Ruzsa participated in the International Mathematical Olympiad
International Mathematical Olympiad
The International Mathematical Olympiad is an annual six-problem, 42-point mathematical olympiad for pre-collegiate 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 in 1980...
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 in 1976. Since then he has been at the Alfréd Rényi Institute of Mathematics
Alfréd Rényi Institute of Mathematics
The Alfréd Rényi Institute of Mathematics 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. Since its creation, the institute has been the center of mathematical research in Hungary. It received...
of the Hungarian Academy of Sciences
Hungarian Academy of Sciences
The Hungarian Academy of Sciences is the most important and prestigious learned society of Hungary. Its seat is at the bank of the Danube in Budapest.-History:...
. He was awarded the Rollo Davidson Prize
Davidson Prize
The Rollo Davidson Prize is a prize awarded annually to early-career probabilists by the Rollo Davidson trustees.- History :In 1970, Rollo Davidson, a Fellow-elect of Churchill College, Cambridge died on Piz Bernina, a mountain in Switzerland...
in 1988. He was elected corresponding member (1998) and member (2004) of the Hungarian Academy of Sciences
Hungarian Academy of Sciences
The Hungarian Academy of Sciences is the most important and prestigious learned society of Hungary. Its seat is at the bank of the Danube in Budapest.-History:...
. He was invited speaker at the European Congress of Mathematics
European Congress of Mathematics
The European Congress of Mathematics is an international congress of the mathematics community, held every four years. Its objectives are "to present various new aspects of pure and applied mathematics to a wide audience, to be a forum for discussion of the relationship between mathematics and...
at Stockholm, 2004, and in the Combinatorics section of the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
in Madrid, 2006.
With Endre Szemerédi
Endre Szemerédi
Endre Szemerédi is a Hungarian mathematician, working in the field of combinatorics and theoretical computer science. He is the State of New Jersey Professor of computer science at Rutgers University since 1986...
he proved that on n points only o(n2) triples can be given such that the union of any 3 of them contains at least 7 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
Freiman's theorem
In mathematics, Freiman's theorem is a combinatorial result in number theory. In a sense it accounts for the approximate structure of sets of integers that contain a high proportion of their internal sums, taken two at a time.The formal statement is:...
. Ruzsa also showed the existence of a Sidon sequence which has at least x0.41 elements up to x.
In a result complementing the Erdős–Fuchs theorem
Erdos–Fuchs theorem
In mathematics, in the area of combinatorial number theory, the Erdős–Fuchs theorem is a statement about the number of ways that numbers can be represented as a sum of two elements of a given set, stating that the average order of this number cannot be close to being a linear function.The theorem...
he showed that there exists a sequence a0, a1, ... of natural numbers such that for every n the number of solutions of the inequality ai + aj ≤ n is cn + O(n1/4log n) for some c > 0.
External links
- Some of Ruzsa's papers at the Rényi Institute