Spectral Sets And Tiles On Local Fields

Let G be a locally compact Abelian group and G be its dual group.Let ?(?)G be a Borel measurable subset of positive and finite Haar measure.We call ? a spectral set if there exists a set A C G of continuous characters of G which forms an orthogonal basis of the space L2(?)of square Haar-integrabl functions.Such a set A is called a spectrum of ? and(?,(?)s called a spectral pair.We say that the set ? tiles G by translation if there exists a set T C G such that the family of sets{? +t}t?T constitutes a partition of G up to measure zero.Such a set T is called a tiling complement or a tiling set of ? and(?,T)is called a tiling pair.For spectral set and tiling,We have the following:Spectral set conjecture:A set ? is a spectral set if and only if ? is a translational tile of G.In this thesis,we focus on the spectral set conjecture on the vector space over the non-Archimedean local fields.The organizations of the thesis are as follows:In Chapter one,we introduce the spectral set conjecture and its research back-grounds.In Chapter two,taking the field Qp of p-adic numbers as an example,we in-troduce the non-Archimedean absolute value on local fields and special topological structure of local fields.Furthermore,we discuss some integration theory on them.In Chapter three,we consider the spectral set conjecture on vector spaces over local fields.More specifically,let ? be a Haar measurable subset of Kd with 0<m(?)<?,and let A be quasi-lattice in Kd and A*be its dual quasi-lattice.Then(?,(?))is tiling pair if and only if(?,(?)*)is pectral pair.The result has been published on J.Math.Anal.Appl.In Chapter four,we discuss the spectral set and tiling problems on the Cartesian products over the field Qp of p-adic numbers.Let ?1(?)Qpd1 and ?2(?)QPd2 be bounded measurable sets,respectively.We consider the following problem:Do the spectralities(resp.tiling properties)of both ?1 and ?2 imply the spectralities(resp.tiling properties)of Cartesian products ?1× ?2 in Qpd1×Qpd2,and vice versa?For tiles,we give a positive answer to the above question.For general spectral sets,if two sets are spectral on the corresponding spaces,respectively.Then the Cartesian product of them is a spectral set on the product space.In Chapter five,we summarize the current work and make some suggestions to the future research.