| Let Ω denote the bounded measurable set on Rd.If there is a countable set J(?)Rd such that holds almost everywhere and for any r,r’∈J,r≠r’,m((Ω+r)(?)(Ω+r’))=0,where m represents the Lebesgue measure,then we call Ω tiles Rd by translating and J a tiling set corresponding to Ω.Similar definable integer tile and corresponding integer tiling set.This thesis mainly prepares for the study of one-dimensional Fuglede’s Conjec-ture,and summarizes some relevant conclusions,specifically the nature of integer tile and self-similar tile when d=1.The main content is divided into the following two parts:The third section studies the properties of integer tile around the conditions(T1)and(T2)proposed by Coven and Meyerowitz,we will show finite discrete set D is an integer tile if it satisfies conditions(T1)and(T2);On the contrary,if the digit set D is an integer tile,it can only be inferred(T1)condition.There is also a spectrum on the finite discrete set D that satisfies the conditions(T1)and(T2).The fourth section mainly describes the form of digit set D that can generate self-similar tile.When D is a standard digit set,the corresponding T(b,D)is a self-similar tile.In addition,when D is a product-form digit set,the corresponding T(b,D)is a self-similar tile.Specifically,it shows that when#D is pn(p is a prime number)and pq(p and q are different prime numbers),T(b,D)is a sufficient and necessary condition for self-similar tile. |
