On Badly Approximable Fractional Parts Of Polynomials

Let α∈R and N>1.Classical Dirichlet Theorem says that there is a natural n≤N with‖nα‖<1/N, where‖y‖=minp∈z|y-p|denotes the distance from y to the nearest integer.Considerable difficulties arise if the linear polynomial f(n)=n is replaced by nonlinear polynomials. Heilbronn’s Theorem on quadratic polynomials f(n)=n2tells us that for any ε>0, there is a n∈N with n≤N and‖n2α‖<N-1/2+ε, according to this result,we haveSchmidt asked if it was true that for any α∈R,In[3],Moshchevitin gave a negative answer to Schmidt’s question,and he proved that for any polynomial f(x)∈R[x],the set had Hausdorff dimension at leas d/d+1,where d was the degree of f(x).Bugeaud and Moshchevitin improved the above result by indicating that the Hausdorff dimension of {α∈R:lim infnâ†'∞nlogn‖f(n)·α‖>0}is1.But they omitted the details of the proof. The aim of this thesis is to provide a detailed proof of Bugeaud and Moshchevitine’s result.