The Study Of Some Problems Related To Littlewood Conjecture

Since two centuries the theory of Diophantine approximation has got a tremendous development. It has been an important branch of the number theory.In this paper we recall some classical results (including Khintchine's theorem and Jarnik's theorem) without proof and give an important concept of badly approximable numbers, meanwhile we associates with the theory of continued fractions, especially discuss the real number with bounded partial quotients. Subsequently, we begin by a survey on the celebrated Littlewood conjecture and focus on summarizing some recent developments on this conjecture. Besides, we introduce some famous problems which are related to the Littlewood conjecture. We recall some well-known results on Hausdorff measure and dimension. Significantly we present a useful theorem which is popularly used in calculating the Hausdorff dimension. The last section is concerned with the theory of uniform distribution modulo one.The main purpose of this paper is to provide new, short and elementary results on the problem which related to Littlewood conjecture. The first result is, for any given real numberαwith bounded partial quotients, we can construct explicitly continuum many real numbersβwith bounded partial quotients for which the pair(α,β)satisfies a strong form of the Littlewood conjecture. Our proof mainly rests on the basic theory of continued fractions. The second result is, we firstly prove the existence of real numbers(α,β)such that liminf 20q→∞(q logq) qαqβ>. Then we give a further result that the set of pairs (α,β) satisfies the above inequality has full Hausdorff dimension. Our method was firstly introduced by Peres and Schlag.