On The Set Of Normal Number Whose Continued Fraction Coefficients Are Bounded

The subject of metric number theory has its origins in the researches of Borel,Weyl,and Khintchine in the first three decades of the twentieth century.Normal number and Diophantine approximation are two fundamental issues of the number theory.In this paper we first recall some classical results of Normal number and Diophantine approximation.For example:the set Bad of badly approximation in[0,1]have a Lebesgue measure of 0 and a Hausdorff dimension of 1;the set S of normal number in[0,1]have a Lebesgue measure of 1.Finally,we study a set of normal number whose continued fraction coefficients are bounded.We define the set K of normal number whose fraction coefficients are bound K={??Bad,? is a normal number}The main purpose of this paper is to prove the set K has a Hausdorff measure of 1.Due to the relation between normal number and uniform distribution,we can transform this question as any positive integer N>1,the sequence {bnx} is uniformly distributed for all x?Bad.Our proof mainly rely on the Davenport,Erdos and LeVeque criterion and the decay of Fourier transform of Kaufman measure on Bad.