Furstenberg gained attention at an early stage in his career for producing an innovative
topological proof of the infinitude of prime numbers in 1955. In a series of articles beginning in 1963 with A
Poisson Formula for Semi-Simple Lie Groups, he continued to establish himself as a ground-breaking thinker. His work showing that the behavior of random walks on a group is intricately related to the structure of the group—which led to what is now called the
Furstenberg boundary—has been hugely influential in the study of lattices and Lie groups. In his 1967 paper,
Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Furstenberg introduced the notion of 'disjointness,' a notion in ergodic systems that is analogous to coprimality for integers. The notion turned out to have applications in areas such as number theory, fractals, signal processing and electrical engineering. In 1977, he gave an ergodic theory reformulation, and subsequently proof, of
Szemerédi's theorem. This is described in his 1977 paper,
Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. Furstenberg used methods from ergodic theory to prove a celebrated result by Endre Szemerédi, which states that any subset of integers with positive upper density contains arbitrarily large arithmetic progressions. His insights then led to later important results, such as the proof by
Ben Green and
Terence Tao that the sequence of prime numbers includes arbitrary large arithmetic progressions. He proved unique ergodicity of horocycle flows on compact hyperbolic
Riemann surfaces in the early 1970s. The
Furstenberg boundary and
Furstenberg compactification of a
locally symmetric space are named after him, as is the
Furstenberg–Sárközy theorem in
additive number theory. ==Personal life==