Hausdorff Dimension Of Some Limsup Sets In Diophantine Approximation

The theoretical development of Diophantine approximation began two centuries ago.It is an important branch of number theory with a long history.The core problem of Diophantine approximation is to study the rational approximation of real numbers.In 1842,a conclusion about metric properties in rational approximation of real numbers was first given,that is,the classical Dirichlet theorem.In 1891,Hurwitz improved the constant of approximation function in Dirichlet theorem and determined the best value for all real numbers.In 1924,Khinchine extended approximation function to more general function,and studied Lebesgue measure of set in Diophantine approximation.In 1931,Jarnik described the scale of the set studied in Khinchine theorem more precisely,that is,the Hausdorff measure of the set,after all,Hausdorff measure is the refinement of Lebesgue measure.However,in the above studies,there are certain monotonicity restrictions on approximation functions.For general approximation functions,Duffin and Schaeffer gave Duffin-Schaeffer conjectures on Lebesgue measures in the 1940 s.Therefore,the famous Khinchine theorem is the general Duffin-Schaeffer conjecture under certain conditions,while for Hausdorff's Duffin-Schaeffer conjecture on measure is given by Beresnevich and Velani.In 1996,Hinokuma and Shiga studied and proved the Hausdorff dimension of sets in Khinchine theorem,But its approximation function is not necessarily monotonic.In this paper,on the basis of Hinokuma and Shiga's research,we combine the restriction conditions of the set in Duffin Schaeffer's conjecture and further strengthen the research on it,that is,we study the coprime of rational numerator and denominator,and the divisor is odd,The denominator is the Hausdorff dimension of the set in Duffin-Schaeffer conjecture in the case of prime number.This will depend strictly on the distribution of such rational numbers.