ON THE LIMIT POINTS OF (anξ)n=1∞ MOD 1 FOR SLOWLY INCREASING INTEGER SEQUENCES (an)n=1∞

In this paper, we are interested in sequences of positive integers (a(n))(n=1)(infinity) such that the sequence of fractional parts {a(n)xi}(n=1)(infinity) has only finitely many limit points for at least one real irrational number.. We prove that, for any sequence of positive numbers (g(n))(n=1)(infinity) satisfying g(n) >= 1 and lim(n ->infinity) g(n) = infinity and any real quadratic algebraic number alpha, there is an increasing sequence of positive integers (a(n))(n=1)(infinity) such that a(n) <= ng(n) for every n epsilon N and lim(n ->infinity){a(n)a} = 0. The above bound on a(n) is best possible in the sense that the condition lim(n ->infinity) g(n) = infinity cannot be replaced by a weaker condition. More precisely, we show that if (a(n))(n=1)(infinity) is an increasing sequence of positive integers satisfying lim inf(n ->infinity) a(n)/n < infinity and xi is a real irrational number, then the sequence of fractional parts {a(n)xi}(n=1)(infinity) has infinitely many limit points.