The aim of this paper is to investigate the size properties of a planar set whose distance set has some prescribed arithmetic combinatorics. Such research is motivated by the conjecture that the disk has no more than ...The aim of this paper is to investigate the size properties of a planar set whose distance set has some prescribed arithmetic combinatorics. Such research is motivated by the conjecture that the disk has no more than 3 orthogonal exponentials. By proving a shifted version of ErdSs-Solymosi's theorem on the distance sets, we give some grounds on the conjecture. The results obtained here extend the corresponding results of Iosevich and Jaming in a simple manner.展开更多
The self-affine measure μM,Dassociated with an iterated function system{φd(x)=M(x + d)}is uniquely determined. It only depends upon an expanding matrix M and a finite digit set D. In the present paper we give some s...The self-affine measure μM,Dassociated with an iterated function system{φd(x)=M(x + d)}is uniquely determined. It only depends upon an expanding matrix M and a finite digit set D. In the present paper we give some sufficient conditions for finite and infinite families of orthogonal exponentials. Such research is necessary to further understanding the non-spectral and spectral of μM,D. As an application,we show that the L~2(μM,D) space has infinite families of orthogonal exponentials on the generalized three Sierpinski gasket. We then consider the spectra of a class of self-affine measures which extends several known conclusions in a simple manner.展开更多
The problems of determining the spectrality or non-spectrality of a measure have been received much attention in recent years. One of the non-spectral problems on <span style="white-space:nowrap;"><...The problems of determining the spectrality or non-spectrality of a measure have been received much attention in recent years. One of the non-spectral problems on <span style="white-space:nowrap;"><em>μ<sub>M,D</sub></em></span><sub> </sub>is to estimate the number of orthogonal exponentials in <em>L</em><sup>2</sup><span style="white-space:normal;">(</span><em>μ<sub>M,D<span style="white-space:normal;">)</span></sub></em>. In the present paper, we establish some relations inside the zero set <img src="Edit_2196df81-d10f-4105-a2a9-779f454a56c3.png" width="55" height="23" alt="" /> by the Fourier transform of the self-affine measure <em>μ<sub>M,D</sub></em>. Based on these facts, we show that <em>μ<sub>M,D</sub></em> is a non-spectral measure<em><sub> </sub></em>and there exist at most 4 mutually orthogonal exponential functions in <em style="white-space:normal;"><em style="white-space:normal;">L</em><sup style="white-space:normal;">2</sup><span style="white-space:normal;">(</span><span style="white-space:normal;"></span><em style="white-space:normal;">μ<sub>M,D)</sub></em></em>, where the number 4 is the best possible. This extends several known conclusions.展开更多
Let M=ρ^(-1)I∈Mn(R)be an expanding matrix with 0<|ρ|<1 and D■Z^(n)be a finite digit set with O∈D and Z(mD)■Z(mD)■Z(mD)∪{0}■m^(-1)z^(n)for a prime m,where Z(mD):=(Ede emi(a)=O),LetμM.D be theassociateds...Let M=ρ^(-1)I∈Mn(R)be an expanding matrix with 0<|ρ|<1 and D■Z^(n)be a finite digit set with O∈D and Z(mD)■Z(mD)■Z(mD)∪{0}■m^(-1)z^(n)for a prime m,where Z(mD):=(Ede emi(a)=O),LetμM.D be theassociatedsel-simiar measure defined by M.DO)-ZaeDμM,D(M()-d).In this paper,the necessary and sufficient conditions for L2(μM,D)to admit infinite orthogonal exponential functions are given.Moreover,by using the order theory of polynomial,we estimate the number of orthogonal exponential functions for all cases that L^(2)(μM,D)does not admit infinite orthogonal exponential functions.展开更多
基金Supported by Key Project of Ministry of Education of China (Grant No. 108117) and National Natural Science Foundation of China (Grant No. 10871123)
文摘The aim of this paper is to investigate the size properties of a planar set whose distance set has some prescribed arithmetic combinatorics. Such research is motivated by the conjecture that the disk has no more than 3 orthogonal exponentials. By proving a shifted version of ErdSs-Solymosi's theorem on the distance sets, we give some grounds on the conjecture. The results obtained here extend the corresponding results of Iosevich and Jaming in a simple manner.
基金supported by National Natural Science Foundation of China(Grant No.11101334)
文摘The self-affine measure μM,Dassociated with an iterated function system{φd(x)=M(x + d)}is uniquely determined. It only depends upon an expanding matrix M and a finite digit set D. In the present paper we give some sufficient conditions for finite and infinite families of orthogonal exponentials. Such research is necessary to further understanding the non-spectral and spectral of μM,D. As an application,we show that the L~2(μM,D) space has infinite families of orthogonal exponentials on the generalized three Sierpinski gasket. We then consider the spectra of a class of self-affine measures which extends several known conclusions in a simple manner.
文摘The problems of determining the spectrality or non-spectrality of a measure have been received much attention in recent years. One of the non-spectral problems on <span style="white-space:nowrap;"><em>μ<sub>M,D</sub></em></span><sub> </sub>is to estimate the number of orthogonal exponentials in <em>L</em><sup>2</sup><span style="white-space:normal;">(</span><em>μ<sub>M,D<span style="white-space:normal;">)</span></sub></em>. In the present paper, we establish some relations inside the zero set <img src="Edit_2196df81-d10f-4105-a2a9-779f454a56c3.png" width="55" height="23" alt="" /> by the Fourier transform of the self-affine measure <em>μ<sub>M,D</sub></em>. Based on these facts, we show that <em>μ<sub>M,D</sub></em> is a non-spectral measure<em><sub> </sub></em>and there exist at most 4 mutually orthogonal exponential functions in <em style="white-space:normal;"><em style="white-space:normal;">L</em><sup style="white-space:normal;">2</sup><span style="white-space:normal;">(</span><span style="white-space:normal;"></span><em style="white-space:normal;">μ<sub>M,D)</sub></em></em>, where the number 4 is the best possible. This extends several known conclusions.
基金Supported by the NNSF of China(Grant Nos.12071125,12001183 and 11831007)the Hunan Provincial NSF(Grant Nos.2020JJ5097 and 2019JJ20012)the SRF of Hunan Provincial Education Department(Grant No.19B117)。
文摘Let M=ρ^(-1)I∈Mn(R)be an expanding matrix with 0<|ρ|<1 and D■Z^(n)be a finite digit set with O∈D and Z(mD)■Z(mD)■Z(mD)∪{0}■m^(-1)z^(n)for a prime m,where Z(mD):=(Ede emi(a)=O),LetμM.D be theassociatedsel-simiar measure defined by M.DO)-ZaeDμM,D(M()-d).In this paper,the necessary and sufficient conditions for L2(μM,D)to admit infinite orthogonal exponential functions are given.Moreover,by using the order theory of polynomial,we estimate the number of orthogonal exponential functions for all cases that L^(2)(μM,D)does not admit infinite orthogonal exponential functions.