### 数学代写|数论作业代写number theory代考|Proofs of Theorems 3.1 and 3.2

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## 数学代写|数论作业代写number theory代考|Proofs of Theorems 3.1 and 3.2

The equivalence classes of edge-equivalent permutations $\sigma \in S_{N}$ for the $N$-tuple $Y$ were introduced in [8]. This definition was given in terms of directed and undirected graphs associated with permutations. We refer to [8, Sect.3] for the details and would like to notice here that the following fact was also proved: two permutations $\sigma, \sigma^{\prime} \in S_{N}$ are edge-equivalent if and only if $V_{\sigma}(Y)=V_{\sigma^{\prime}}(Y)$ for every $Y \in \mathbb{A}$. We recall that $V_{\sigma}$ is defined in (2.6) (it is the the metric length of the aforementioned directed metric graph associated with $\sigma$ and $Y$ ).

Let us denote by $\tilde{n} \in \mathbb{N}$ the number of edge-equivalence classes in $S_{N}$ and let us take one representative $\tilde{\sigma}{j}, j=1, \ldots, \tilde{n}$, in each of them. We use also the following observation of [8]: if $Y$ belongs to the set $$\mathbb{A}{1}:=\left{Y \in \mathbb{A}: V_{\sigma_{j}}(Y) \neq V_{\widetilde{\sigma}{m}}(Y) \text { if } j \neq m\right},$$ then there is no cancellation of the exponential monomials (2.5) with the highest possible frequency $V(Y)$ after the summation of (2.5) required by the Leibniz formula for det $\Gamma{Y}$. Thus, for every $Y \in \mathbb{A}_{1}$ the Weyl-type asymptotics takes place.

Proof of Theorem $3.2$ For each permutation $\sigma$, the function $V_{\sigma}(\cdot)$ is a sum of terms of the form
$$\left|Y_{j}-Y_{j^{\prime}}\right|=\left(\sum_{m=1}^{3}\left[Y_{j, m}-Y_{j^{\prime}, m}\right]^{2}\right)^{1 / 2}$$
where $j^{\prime}=\sigma(j)$ and where $Y_{j, m}, m=1,2,3$, are the $\mathbb{R}^{3}$-coordinates of $Y_{j}, 1 \leq$ $j \leq N$. Therefore, $V_{\sigma}(\cdot)$ is a real analytic function in the variables $Y_{j, m}(1 \leq j \leq N$, $m=1,2,3$ ) on $\mathbb{A}$.

Let us take now the representatives $\tilde{\sigma}{j}, j=1, \ldots, \tilde{n}$, of edge-equivalent classes of permutations, which were described above. We see that the function $f{j, m}(Y)=$ $V_{\widetilde{\sigma}{j}}(Y)-V{\widetilde{\sigma}{m}}(Y)$ is real analytic on $\mathbb{A}$. Moreover, if $j \neq m$ this function is not trivial on $\mathbb{A}$, and so the set $\mathbb{A}{0}^{j, m}:=\left{Y \in \mathbb{A}: f_{j, m}(Y)=0\right}$ of its zeroes is a proper analytic subset of $\mathbb{A}$.
We will use the following well-known fact (see e.g., [42]):

a proper analytic subset of an open set in $\mathbb{R}^{d}$ has measure zero.
Thus, each of the sets $\mathbb{A}{0}^{j, m}$ with $j \neq m$ is of measure zero and so is their union $\tilde{\mathbb{A}}{0}=\bigcup_{1<j<m<\tilde{n}} \mathbb{A}{0}^{j, m}$. This union is obviously also a proper analytic subset of $\mathbb{A}$ (it is the set of zeros of the function $\left.f(Y)=\prod{1<j<m \leq \tilde{n}} f_{j, m}\right)$.

As it was mentioned above, the results of [8] imply that, if $Y \in \mathbb{A} \backslash \widetilde{\mathbb{A}}{0}$, the Weyltype asymptotics takes place. Summarizing, we see that $\mathbb{A}{0} \subset \widetilde{\mathbb{A}}{0}$, and that $\widetilde{\mathbb{A}}{0}$ is a proper analytic subset of $\mathbb{A}$ and it has measure zero. This completes the proof.

## 数学代写|数论作业代写number theory代考|Point Process Describing the Asymptotics of Random

The goal of this section is to show that the structure of the set of random resonances of $H_{\Upsilon}$ near $\infty$ can be described by a point process on $\mathbb{R}{+}$. Consider first a deterministic collection $Y \subset \mathbb{R}^{3}$ such that $Y$ is simple and $2 \leq # Y<\infty$. (4.1) Then the multiset of resonances $\Sigma\left(H{Y}\right)$ has the global structure of a finite number of sequences going to $\infty$ with prescribed asymptotics [7]. Namely, there exists a sequence $\left{K_{j}(Y)\right}_{j=1}^{n_{1}(Y)}$ of $n_{1}(Y) \in \mathbb{N}$ positive numbers such that the multiset $\Sigma\left(H_{Y}\right)$ is essentially the union $\bigcup_{j=1}^{N}\left{k_{j, m}\right}_{m \in \mathbb{Z}}$ of the sequences satisfying
$$k_{j, m}=\pi K_{j}(Y)(2 m+1)-i K_{j}(Y) \operatorname{Ln}\left|\pi K_{j}(Y)(2 m+1)\right|+O(1) \text { as }|m| \rightarrow \infty,$$
$j=1, \ldots, n_{1}(Y)$. (In [7] a more precise asymptotic formula is given, but we do not need it in the present paper.) ‘Essentially’ in this context means that one multiset can be obtained from the other by possible addition or exclusion of a finite number of elements.

The collection $K(Y)=\left{K_{j}(Y)\right}_{j=1}^{n_{1}(Y)}$ of the leading parameters of the asymptotic sequences (4.2) can be considered as a multiset and we assume that $K_{j}$ are ordered such that
$$K_{1}(Y) \leq K_{2}(Y) \leq \cdots \leq K_{n_{1}(Y)}(Y)$$
Note that some of its elements are actually multiple. Namely, the following facts were proved in [7] under condition (4.1):
\begin{aligned} 2 \leq n_{1}(Y) &=# K(Y)<# Y, \ 1 / \operatorname{diam} Y &=K_{1}(Y)=K_{2}(Y) \end{aligned}
(the latter implies that the multiplicity $\operatorname{mult}\left(K_{1}(Y)\right.$ ) of the minimal parameter $K_{1}(Y)$ is at least 2).

## 数学代写|数论作业代写number theory代考|Limits of Random Asymptotic Structures Under

As it is shown in Sect. 4, the asymptotic behaviour of random resonances at $\infty$ is described by the finite point process $K(\Upsilon)=\left{K_{j}(\Upsilon)\right}_{j=1}^{# K(\Upsilon)}$ on $\mathbb{R}$. This naturally poses a question about the asymptotics of the random counting measures $\eta_{K(\Upsilon[m])}$ for a reasonably chosen sequence of point processes $\Upsilon^{[m]}$. It makes sense to assume thăt with the growth of $m \rightarrow \infty$ ẻithere the ‘intennsity’ or thé sup̄port of $\Upsilon^{[m]}$ grows unboundedly.

As a simple example of such a reasonable sequence of point processes, one can take uniform binomial processes $\Upsilon^{[m]} \in \Theta\left(m, \mathbb{B}{r}\right)$ in a fixed $3-\mathrm{D}$ ball $\mathbb{B}{r}$ with the total intensity $m$ going to $+\infty$. That is, for each $m$ there exists a sequence $\left{\xi_{j}^{[m]}\right}_{j=1}^{m}$ of independent random variables with uniform distribution in the unit ball $\mathbb{B}{r}$ such that $\Upsilon^{[m]}=\left{\xi{j}^{[m]}\right}_{j=1}^{m}$. Formally, the elements $\xi_{j}^{[m]}$ of the sequence $\Upsilon^{[m]}$ depend on $m$. However, this dependence can be often neglected. There exists an infinite sequence $\left{\xi_{j}\right}_{j=1}^{\infty}$ of uniformly distributed in $\mathbb{B}{r}$ i.i.d. random variables such that each of the point processes $$\tilde{\Upsilon}^{[m]}=\left{\xi{j}\right}_{j=1}^{m}, \quad m \in \mathbb{N},$$
has the same distribution as $\Upsilon^{[m]}$. For all the purposes of the present paper, $\left{\Upsilon^{[m]}\right}_{1}^{\infty}$ can be replaced by $\left{\tilde{\Upsilon}^{[m]}\right}_{1}^{\infty}$ (the only reason for avoiding this replacement is to simplify the notation).

The first questions in connection with the limiting behavior of $K\left(\Upsilon^{[m]}\right)$ concern the limits of the random variables
$n_{1}^{[m]}:=n_{1}\left(\Upsilon^{[m]}\right)=# K\left(\Upsilon^{[m]}\right), \quad \mathcal{K}{\min }^{[m]}:=\mathcal{K}{1}\left(\Upsilon^{[m]}\right)$, and $\quad \mathcal{K}{\max }^{[m]}:=\mathcal{K}{n_{1}^{[m]}}^{[m]}\left(\Upsilon^{[m]}\right)$
Another interesting limiting behavior question concerns the total asymptotic densities $\operatorname{Ad}\left(H_{\Upsilon(m])}\right)$ (cf. the introduction to $\left.[51]\right)$. Recall that $V(Y):=\max {\sigma \in S{n Y}}$ $\sum_{j=1}^{# Y}\left|Y_{j}-Y_{\sigma(j)}\right|$ is called the size of the set $Y$ (see Sect. 3).

## 数学代写|数论作业代写number theory代考|Proofs of Theorems 3.1 and 3.2

\mathbb{A}{1}:=\left{Y \in \mathbb{A}: V_{\sigma_{j}}(Y) \neq V_{\widetilde{\sigma}{m}}(Y) \text { 如果 } j \neq m\right}，\mathbb{A}{1}:=\left{Y \in \mathbb{A}: V_{\sigma_{j}}(Y) \neq V_{\widetilde{\sigma}{m}}(Y) \text { 如果 } j \neq m\right}，那么没有对最高可能频率的指数单项式（2.5）的抵消在(是)在对 det 的莱布尼茨公式要求的 (2.5) 求和之后Γ是. 因此，对于每个是∈一种1发生 Weyl 型渐近线。

|是j−是j′|=(∑米=13[是j,米−是j′,米]2)1/2

## 数学代写|数论作业代写number theory代考|Point Process Describing the Asymptotics of Random

ķj,米=圆周率ķj(是)(2米+1)−一世ķj(是)ln⁡|圆周率ķj(是)(2米+1)|+这(1) 作为 |米|→∞,
j=1,…,n1(是). （在 [7] 中给出了更精确的渐近公式，但我们在本文中不需要它。）在这种情况下，“本质上”意味着可以通过可能添加或排除有限数从另一个多重集获得的元素。

ķ1(是)≤ķ2(是)≤⋯≤ķn1(是)(是)

\begin{aligned} 2 \leq n_{1}(Y) &=# K(Y)<# Y, \ 1 / \operatorname{diam} Y &=K_{1}(Y)=K_{2}( Y) \end{对齐}\begin{aligned} 2 \leq n_{1}(Y) &=# K(Y)<# Y, \ 1 / \operatorname{diam} Y &=K_{1}(Y)=K_{2}( Y) \end{对齐}
（后者意味着多重性很多⁡(ķ1(是)) 的最小参数ķ1(是)至少为 2)。

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