He studied under A. A. Markov at the University of St. Petersburg, graduating in 1912. He then began research in probability theory. He converted to the Russian Orthodox religion on marrying in 1916. He was appointed professor at the University of Perm in 1917, and was caught up in the Civil War over the next two years. In 1920 he took a position in Petrograd University.
In 1924 he went to Copenhagen and Harald Bohr, on a Rockefeller Fellowship, where he worked on almost periodic functions. After a visit to G.H. Hardy in Oxford, he had appointments at Liverpool University in 1926, and the University of Cambridge in 1927.
He worked mainly on combinatorial methods and questions in real analysis, such as the Kakeya needle problem, and Hausdorff dimension (often now called Hausdorff-Besicovitch dimension). These two particular areas have proved increasingly important as the years have gone by.
He was a major influence on the economist Piero Sraffa, after 1940, when they were both Fellows of Trinity College.
He was J.E. Littlewood's successor in 1950 in the Rouse Ball chair at Cambridge, retiring in 1958. He died in Cambridge.