In 1917 Perm became an independent institution (it later became the University of Moscow) and in that year Besicovitch was appointed professor of mathematics there. Perm suffered badly in the troubles of 1919 despite the best efforts of Besicovitch to protect the books and other university property. In the following year he accepted a chair back at St Petersburg (by then called Leningrad).
Besicovitch left Russia for Copenhagen in 1924 and worked with Harald Bohr. He had applied for permission to work abroad but it had been refused so he escaped across the border with a colleague under the cover of darkness. He managed to reach Copenhagen where he was supported for a year with a Rockefeller Fellowship.
After a visit to Oxford in 1925 Hardy found a post for him in Liverpool. Then in 1927 he moved to Cambridge. His wife remained in Russia and the marriage was dissolved in 1928. Two years later Bessy, as he was known, married the 16 year old daughter of a previous girlfriend.
In 1950 he succeeded Littlewood to the Rouse Ball Chair of mathematics at Cambridge. He held this chair until he retired in 1958. For eight successive years following his retirement he visited different universities in the USA. He then returned to Trinity College, Cambridge where he spent in total over 40 years of his life.
His work on sets of non-integer dimension was an early contribution to fractal geometry. Hausdorff, in 1918, had extended Carathéodory's theory of measure to sets having finite measure of non-integral order. Besicovitch, around 1930, extended his density properties of sets to those of finite Hausdorff measure.
Besicovitch was famous for his work on almost periodic functions, an interest in which came from his time in Copenhagen with Harald Bohr. He wrote an influential text covering his work in this area Almost periodic function in 1932.
One of the achievements, with which he will always be associated, was his solution of the Kakeya problem on minimising areas. The problem had been posed in 1917 by a Japanese mathematician S Kakeya and asked what was the smallest area in which a line segment of unit length could be rotated through 2p. Besicovitch proved in 1925 that given any e, an area of less than e could be found in which the rotation was possible. The figures that resulted from Besicovitch's construction were highly complicated, unbounded figures.
In  Burkill describes Besicovitch's work in these terms:-
At one time or another he shed new light on many aspects of the classical theory of real functions. He was more likely than anyone else to solve a problem which had seemed intractable, commonly the solution needed, by way of proof or counter-example, an ingenious and intricate construction. This work was more congenial to him than the abstract developments of, for example, functional analysis.Besicovitch received many honours for his work. He received the Adams Prize from the University of Cambridge in 1930 for his work on almost periodic functions. He was elected a fellow of the Royal Society in 1934, and in 1952 received the Sylvester Medal from the Society:-
... in recognition of his outstanding work on almost-periodic functions, the theory of measure and integration and many other topics of the theory of functions.In 1950 he was awarded the De Morgan Medal from the London Mathematical Society.
Burkill describes his personal characteristics in  saying:-
His intellectual gifts were matched by his generous sympathy which endeared him to pupils, colleagues and a wide circle of friends.
List of References
Article by: J.J. O'Connor and E.F. Robertson
Source: MacTutor History of Mathematics archive