The Hausdorff Dimension Of The Set Of Points With The Same Engel And Sylvester Expansions

This paper mainly introduces the definition of Engel expansion and Sylvester expansion in number theory and their related metric properties,and emphasizes the Hausdorff dimension of the set and generalized set with the same Engel and Sylvester expansions.Chapter 1,Introduction,it mainly introduces the background and significance of the problem.Similar to the number of decimal expansions,the Engel expansion and the Sylvester expansion are the other two infinite series representations of numbers.As early as 1971,J.Galambos[1-3] and P.Erd?s[4] systematically studied Engel and Sylvester expansion.They proved the metric properties of the Engel and Sylvester series.C.M.Goldie[5],J.Wu[6-8],Y.Y.Liu[9,10],B.W.Wang[11-13] researched some exceptional sets of Engel and Sylvester expansions.A number of mathematicians are interested in the set E with the same Engel and Sylvester expansions.It is an interesting question to investigate that how large E can be.That is,whether E has positive or zero Lebesgue measure and when its Lebesgue measure is zero,to evaluate its Hausdorff dimension.In [4],J.Galambos[14-16] showed that E is of Lebesgue measure 0.How about the Hausdorff dimension of E? It is subtle and J.Galambos posed it as an unsolved problem.J.Wu[17] studied this set,and proved that the Hausdorff dimension of the set E is 1?2.This article further studies the more general form of Set E,and give the Hausdorff dimension about it.Chapter 2,Preliminary Knowledge,we introduce the definition of Engel's expansion and Sylvester's expansion,the related metric properties,and the Hausdorff measure and dimension.It also provides the proof that the Hausdorff dimension of the set M is 1??.The third part mainly proves that the Hausdorff dimension of the generalized set M with the same set of Engel and Sylvester expansions is 1??.It is divided into upper bound estimation and lower bound estimation.On the one hand,the upper bound for the Hausdorff dimension of M is given by constructing the natural coverage of M,on the other hand,by skillfully constructing the subset G of set M and using the mass distribution principle,we prove that the lower bound of the Hausdorff dimension of the set G is 1??.Thus,it is proved that the lower bound of the Hausdorff dimension of the set M is1??.