Some Problems of Diophantine Approximation by G. H. Hardy and J. E. Littlewood

Let 0 be an irrational number, and a any number between 0 and 1 (0 included). Then it is well known that it is possible to find a sequence of positive integers wa as r —» oo . Now let f(n) denote a positive increasing function of n, integral when n is integral, such as n, n, n\ ..., 2, S, ...,n\, 2\ ..., 2", ..., and let fr denote the value of f(n) for n = nr. The result just stated suggests the following question, which seems to be of considerable interest :—For what forms of f(n) is it true that, for any irrational value of 0, and any value of a such that 0 — a < 1, a sequence nr can be found stich that