ON SEQUENCES (anξ)n≥1 CONVERGING MODULO 1

We prove that, for any sequence of positive real numbers (g(n))(n >= 1) satisfying g(n) >= 1 for n >= 1 and lim(n ->+infinity) g(n) = +infinity, for any real number theta in [0,1] and any irrational real number, there exists an increasing sequence of positive integers (a(n))(n >= 1) satisfying a(n) <= ng(n) for n >= 1 and such that the sequence of fractional parts ({a(n)xi})(n >= 1) tends to theta as it tends to infinity. This result is best possible in the sense that the condition lim(n ->+infinity) g(n) = +infinity cannot, be weakened, as recently proved by Dubickas.