Research On Several Problems In The Diophantine Approximation With Restrictions

We are motivated by the approximation problem under the?-dynamical system solved by Philipp in1967.This article mainly discusses restricted Diophantine approximation problem on R^d.We obtain some Lebesgue measures properties and Hausdorff measures properties about several lim sup sets on R^d.The first chapter introduced the background of this article,and in the second chapter,We have given the necessary preliminaries.Furthermore,we use two chapters to discuss the problem of non-homogeneous approximation of single approximation functions with restricted sets on R^dand the homogeneous approx-imation problem of multi approximation functions with restricted sets.In Chapter Three,we prove for fixed approximation function?,and any positive real sequences{q_k}_k?1which satisfy q_k+1/q_k?1+?,the setW?_A(?):={X=(x₁,x₂,···,x_n)?[0,1)^d:?qX-Y?<?(q),for infinitely many n?N},for any Y?[0,1)^d has 0-1 law under Lebesgue measure and Hausdorff measure.At the same time,we obtain the inhomogeneous Khintchine-Jarník theorem as a corollary.In Chapter Four,we consider the homogeneous problem of multiple approximation functions on R^d,which beyond the conclusion of the ubiquitous system.Furthermore,we obtain the setW_A(?_t)^d_t=1:={X=(x₁,x₂,···,x_d):?q_n·x_t?<?_t(q_n),1?t?d,for infinitely many n?N},has 0-1 law under Lebesgue measure by Reverse Fatou lemma as the primary tool.Finally,we determine the Hausdorff dimension for lim sup set W_A(?_t)^d_t=1.