论文标题
连续数字的Moran措施的非光谱
Non-spectrality of Moran measures with consecutive digits
论文作者
论文摘要
令$ρ=（\ frac {p} {q}）^{\ frac {1} {r}}} <1 $，对于某些$ p，q，q，r \ in \ mathbb {n} $，带有$（p，q）= 1 $和$ \ Mathcal {d} _ {n} = \ {0,1，\ cdot \ cdot \ cdot \ cdot，n_ {n} -1 \} $，其中$ n_ {n {n} $ as PRIME是所有$ n \ in \ Mathbb {n} $，并且是指$ m = \ sup \ {n_ {n}：n = 1,2,3，\ ldots \} <\ infty $。相关的Borel概率度量$μ_{ρ，\ {\ Mathcal {D} _ {n} \}}} =δ__{ρ\ Mathcal {d} _ {1}}}*δ___{ρ ^{2} \ Mathcal {d} _ {2}}*δ_{ρ^{3} \ Mathcal {d} _ {3}}}*\ cdots $$被称为Moran措施。最近，Deng和Li证明了$μ_{ρ，\ {\ Mathcal {d} _ {n} \}}} $是光谱度量，并且仅当且仅当$ \ frac {1} {n_ {n_ {n_ {n}ρ} $是所有$ n \ geq geq 2 $的integer。在本文中，我们证明，如果$ l^{2}（μ_{ρ，\ {\ Mathcal {d} _ {n} \}}）$包含一个无限正交指数集，则存在无限的正整数，然后存在无限的正整数$ n_ _ {l_ {l} $（n _ $（q，q，q，q，n _}}。相反，如果$（q，n_ {n}）= 1 $和$（p，n_ {n}）= 1 $ for \ mathbb {n} $中的所有$ n \，那么最多有$ m $互惠正交指数函数$ l^{2}（2}（2}（μ__）， \ {\ Mathcal {d} _ {n} \}}）$和$ m $是最好的。如果$（q，n_ {n}）= 1 $和$（p，n_ {n}）> 1 $ in \ in \ mathbb {n} $中的所有$ n \，则在$ l^{2}（μ_{μ_{p，\ {p，\ altacal} n} $ l^{2}中有任何数量的正交指数函数。
Let $ρ=(\frac{p}{q})^{\frac{1}{r}}<1$ for some $p,q,r\in\mathbb{N}$ with $(p,q)=1$ and $\mathcal{D}_{n}=\{0,1,\cdot\cdot\cdot,N_{n}-1\}$, where $N_{n}$ is prime for all $n\in\mathbb{N}$, and denote $M=\sup\{N_{n}:n=1,2,3,\ldots\}<\infty$. The associated Borel probability measure $$μ_{ρ,\{\mathcal{D}_{n}\}}=δ_{ρ\mathcal{D}_{1}}*δ_{ρ^{2}\mathcal{D}_{2}}*δ_{ρ^{3}\mathcal{D}_{3}}*\cdots$$ is called a Moran measure. Recently, Deng and Li proved that $μ_{ρ,\{\mathcal{D}_{n}\}}$ is a spectral measure if and only if $\frac{1}{N_{n}ρ}$ is an integer for all $n\geq 2$. In this paper, we prove that if $L^{2}(μ_{ρ, \{\mathcal{D}_{n}\}})$ contains an infinite orthogonal exponential set, then there exist infinite positive integers $n_{l}$ such that $(q,N_{n_{l}})>1$. Contrastly, if $(q,N_{n})=1$ and $(p,N_{n})=1$ for all $n\in\mathbb{N}$, then there are at most $M$ mutually orthogonal exponential functions in $L^{2}(μ_{ρ, \{\mathcal{D}_{n}\}})$ and $M$ is the best possible. If $(q,N_{n})=1$ and $(p,N_{n})>1$ for all $n\in\mathbb{N}$, then there are any number of orthogonal exponential functions in $L^{2}(μ_{ρ, \{\mathcal{D}_{n}\}})$.
