# Extended Watson-Harkins Sum

Robert Reynolds  Department of Mathematics and Statistics, York University,
Toronto, ON, Canada, M3J1P3<EMAIL_ADDRESS>
###### Abstract.

The Watson-Harkins sum involving the product of the cosine and cosecant
functions is extended to derive the finite sum of generalized Hurwitz-Lerch
Zeta functions is derived in terms of the Hurwitz-Lerch Zeta function. A
transformation formula arises for various finite values of the parameters
involved. The finite product of trigonometric functions are also derived. All
the results in this work are new.

###### Key words and phrases:

Hurwitz-Lerch Zeta function, Cauchy integral, finite product

###### 2020 Mathematics Subject Classification:

Primary 30E20, 33-01, 33-03, 33-04

##  1\. Introduction

The work done on theories of the structure of the atom by Watson [1] involved
the finite sum of the cosecant function which was used in the calculation of
the angular velocity of the system. Harkins et al. [2] also produced work on
the structure of the atom where the finite sum of the product the cosine and
cosecant functions were used to calculate the Nicholson period of vibration.
Since the finite sum of trigonometric functions are of high usage and
importance the authors embark on using their method to derive a finite sum of
the generalized cases of trigonometric functions namely the Hurwitz-Lerch Zeta
function.

In this present work we derive the finite sum given by

(1.1) $\sum_{p=0}^{n-1}e^{-\frac{2ij(\pi
lp+mn)}{n}}\left(\Phi\left(e^{\frac{2i(mn+lp\pi)}{n}},-k,-j-\frac{1}{2}i\log(a)\right)\right.\\\
\left.+e^{2i(2j+1)\left(\frac{\pi
lp}{n}+m\right)}\Phi\left(e^{\frac{2i(mn+lp\pi)}{n}},-k,j-\frac{1}{2}i\log(a)+1\right)\right)\\\
\\\
=(2i)^{-k}n\left(\log^{k}(a)+2^{k+1}(in)^{k}e^{2imn}\Phi\left(e^{2imn},-k,1-\frac{i\log(a)}{2n}\right)\right)$

where the variables $k,a,m$ are general complex numbers and
$j=0,1,\ldots,n-1$; $l$ and $n$ are relatively prime numbers. This new
expression is then used to derive special cases in terms of trigonometric
functions.

Some special cases evaluated are in terms of the log-gamma function and
Glaisher’s constant $A$ which is featured in Barnes’ $G$ -Function in section
(5.17) in [3] and Bernoulli number in section (24.2) in [3]. New functional
identities are also derived which are used in the study of the distribution of
prime numbers see section (25.16(i)) in [3], the evaluation of Euler sums see
section (25.16(ii)) in [3]. According to section (25.17) in [3], the zeta
function occurs in the evaluation of the partition function of ideal quantum
gases (both Bose–Einstein and Fermi–Dirac cases), and it determines the
critical gas temperature and density for the Bose–Einstein condensation phase
transition in a dilute gas (Lifshitz and Pitaevskiĭ [4]). Quantum field theory
often encounters formally divergent sums that need to be evaluated by a
process of regularization: for example, the energy of the electromagnetic
vacuum in a confined space (Casimir–Polder effect). It has been found possible
to perform such regularizations by equating the divergent sums to zeta
functions and associated functions (Elizalde [5]).

Our preliminaries start with the application of a contour integral method to a
finite sum. Let $a$, $k$, $m$ and $w$ be general complex numbers and
$n\in\mathbb{Z^{+}}$ and $l$ is a relative prime number, the contour integral
form [6], of the finite sum given by equation (4.4.6.12) in [7] is given by

(1.2) $\frac{1}{2\pi
i}\int_{C}\sum_{p=0}^{n-1}a^{w}w^{-k-1}\csc\left(\frac{\pi
lp}{n}+m+w\right)\cos\left((2j+1)\left(\frac{\pi lp}{n}+m+w\right)\right)dw\\\
=\frac{1}{2\pi i}\int_{C}na^{w}w^{-k-1}\cot(n(m+w))dw$

The derivations follow the method used by us in [6]. This method involves
using a form of the generalized Cauchy’s integral formula given by

(1.3) $\frac{y^{k}}{\Gamma(k+1)}=\frac{1}{2\pi
i}\int_{C}\frac{e^{wy}}{w^{k+1}}dw,$

where $y,w\in\mathbb{C}$ and $C$ is in general an open contour in the complex
plane where the bilinear concomitant [6] has the same value at the end points
of the contour. This method involves using a form of equation (1.3) then
multiplies both sides by a function, then takes the finite sum of both sides.
This yields a finite sum in terms of a contour integral. Then we multiply both
sides of equation (1.3) by another function and take the infinite sum of both
sides such that the contour integral of both equations are the same.

##  2\. The Hurwitz-Lerch Zeta Function

We use equation (1.11.3) in [8] where $\Phi(z,s,v)$ is the Lerch function
which is a generalization of the Hurwitz Zeta $\zeta(s,v)$ and Polylogarithm
functions $Li_{n}(z)$. The Lerch function has a series representation given by

(2.1) $\Phi(z,s,v)=\sum_{n=0}^{\infty}(v+n)^{-s}z^{n}$

where $|z|<1,v\neq 0,-1,-2,-3,..,$ and is continued analytically by its
integral representation given by

(2.2)
$\Phi(z,s,v)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-vt}}{1-ze^{-t}}dt=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-(v-1)t}}{e^{t}-z}dt$

where $Re(v)>0$, and either $|z|\leq 1,z\neq 1,Re(s)>0$, or $z=1,Re(s)>1$.

##  3\. Contour Integral Representation For The Finite Sum Of The Hurwitz-
Lerch Zeta Functions

We use the method in [6]. The cut and contour are in the first quadrant of the
complex $w$-plane with $0<Re(w+m)$. The cut approaches the origin from the
interior of the first quadrant and goes to infinity vertically and the contour
goes round the origin with zero radius and is on opposite sides of the cut.
Using a generalization of Cauchy’s integral formula (1.3) we first replace $y$
by $\log(a)+ix+y$ then multiply both sides by $e^{itx}$ then form a second
equation by replacing $x$ by $-x$ and adding both equations to get

(3.1)
$\frac{e^{-itx}\left(e^{2itx}(\log(a)+ix+y)^{k}+(\log(a)-ix+y)^{k}\right)}{\Gamma(k+1)}\\\
=\frac{1}{2\pi i}\int_{C}2a^{w}w^{-k-1}e^{wy}\cos(x(t+w))dw$

Next we replace $y$ by $i(2y+1)$, $t$ by $\frac{\pi lp}{n}+m$, $x$ by $2j+1$
and multiply both sides by $e^{i(2y+1)\left(\frac{\pi lp}{n}+m\right)}$ and
take the infinite and finite sums over $y\in[0,\infty)$ and $p\in[0,n-1]$,
respectively and simplify in terms of the Hurwitz-Lerch Zeta function to get

(3.2) $-\frac{1}{\Gamma(k+1)}i^{k+1}2^{k}\sum_{p=0}^{n-1}e^{-\frac{2ij(\pi
lp+mn)}{n}}\left(\Phi\left(e^{\frac{2i(mn+lp\pi)}{n}},-k,-j-\frac{1}{2}i\log(a)\right)\right.\\\
\left.+e^{2i(2j+1)\left(\frac{\pi
lp}{n}+m\right)}\Phi\left(e^{\frac{2i(mn+lp\pi)}{n}},-k,j-\frac{1}{2}i\log(a)+1\right)\right)\\\
\\\ =-\frac{1}{2\pi
i}\sum_{y=0}^{\infty}\sum_{p=0}^{n-1}\int_{C}2ia^{w}w^{-k-1}e^{\frac{i(2y+1)(\pi
lp+n(m+w))}{n}}\cos\left((2j+1)\left(\frac{\pi lp}{n}+m+w\right)\right)dw\\\
\\\ =-\frac{1}{2\pi
i}\sum_{p=0}^{n-1}\int_{C}\sum_{y=0}^{\infty}2ia^{w}w^{-k-1}e^{\frac{i(2y+1)(\pi
lp+n(m+w))}{n}}\cos\left((2j+1)\left(\frac{\pi lp}{n}+m+w\right)\right)dw\\\
\\\ =\frac{1}{2\pi i}\int_{C}\sum_{p=0}^{n-1}a^{w}w^{-k-1}\csc\left(\frac{\pi
lp}{n}+m+w\right)\cos\left((2j+1)\left(\frac{\pi lp}{n}+m+w\right)\right)dw\\\
\\\ =\frac{1}{2\pi i}\int_{C}na^{w}w^{-k-1}\cot(n(m+w))dw$

from equation (4.4.6.12) in [7] where $Im(n(m+w))>0$ and
$Re(n(m+w))>0,j=0,\ldots,n-1$; $l$ and $n$ are relatively prime numbers. We
apply Tonelli’s theorem for multiple sums, see page 177 in [9] as the summands
are of bounded measure over the space $\mathbb{C}\times[0,n]$.

##  4\. Contour Integral Representation For The Hurwitz-Lerch Zeta Function

We use the method in [6]. Using equation (1.3) we first replace
$\log(a)+2in(y+1)$ and multiply both sides by $-2ine^{2imn(y+1)}$ then take
the infinite sum over $y\in[0,\infty)$ and simplify in terms of the Hurwitz-
Lerch Zeta function to get

(4.1)
$-\frac{2^{k+1}(in)^{k+1}e^{2imn}\Phi\left(e^{2imn},-k,1-\frac{i\log(a)}{2n}\right)}{\Gamma(k+1)}\\\
\\\ =-\frac{1}{2\pi
i}\sum_{y=0}^{\infty}\int_{C}2ina^{w}w^{-k-1}e^{2in(y+1)(m+w)}dw\\\ \\\
=-\frac{1}{2\pi
i}\int_{C}\sum_{y=0}^{\infty}2ina^{w}w^{-k-1}e^{2in(y+1)(m+w)}dw\\\ \\\
=\frac{1}{2\pi i}\int_{C}na^{w}w^{-k-1}\cot(n(m+w))+ina^{w}w^{-k-1}dw$

from equation (1.232.1) in [10] where the $Im(n(m+w))>0$ in order for the sum
to converge.

###  4.1. The additional contour

We use the method in [6]. Using equation (1.3) we replace $y$ by $\log(a)$ and
multiply both sides by $ni$ and simplify to get

(4.2) $\frac{in\log^{k}(a)}{\Gamma(k+1)}=\frac{1}{2\pi
i}\int_{C}ina^{w}w^{-k-1}dw$

##  5\. The finite sum of Hurwitz-Lerch Zeta functions in terms of the
Hurwitz-Lerch Zeta function

######  Theorem 5.1.

For all $k,a,m\in\mathbb{C}$, $l$ is prime, $j<n-1$ then,

(5.1) $\sum_{p=0}^{n-1}e^{-\frac{2ij(\pi
lp+mn)}{n}}\left(\Phi\left(e^{\frac{2i(mn+lp\pi)}{n}},-k,-j-\frac{1}{2}i\log(a)\right)\right.\\\
\left.+e^{2i(2j+1)\left(\frac{\pi
lp}{n}+m\right)}\Phi\left(e^{\frac{2i(mn+lp\pi)}{n}},-k,j-\frac{1}{2}i\log(a)+1\right)\right)\\\
\\\
=(2i)^{-k}n\left(\log^{k}(a)+2^{k+1}(in)^{k}e^{2imn}\Phi\left(e^{2imn},-k,1-\frac{i\log(a)}{2n}\right)\right)$

###### Proof.

The right-hand sides of relation (3.2), and the addition of relations (4.2)
and (4.1) are identical; hence, the left-hand sides of the same are identical
too. Simplifying with the Gamma function yields the desired conclusion. ∎

######  Example 5.2.

The Degenerate Case.

(5.2) $\sum_{p=0}^{n-1}\csc\left(\frac{\pi
lp}{n}+m\right)\left(-\cos\left((2j+1)\left(\frac{\pi
lp}{n}+m\right)\right)\right)=-n\cot(mn)$

###### Proof.

Use Equation (5.1) and set $k=0$ and simplify using entry (2) in Table below
(64:12:7) in [11]. ∎

##  6\. Trigonometric Product-Recursion Identities And Transformation Formula

Finite trigonometric products are important in many areas of mathematics and
their uses and properties are detailed in [12] page 201. In this section we
will evaluate equation (5.1) for various parameter values and derive new
finite trigonometric products.

######  Example 6.1.

A Finite Product Involving The Exponential Of Trigonometric Functions

(6.1) $\prod_{p=0}^{n-1}\sin^{3}\left(\frac{\pi
p}{n}+\frac{x}{2}\right)\csc^{2}\left(\frac{\pi
p}{n}+\frac{x}{4}\right)\csc\left(\frac{\pi p}{n}+x\right)\\\
\exp\left(4\sin^{2}\left(\frac{x}{4}\right)\left(2\cos\left(\frac{2\pi
p}{n}+x\right)+\cos\left(\frac{2\pi p}{n}+\frac{3x}{2}\right)\right)\right)\\\
\\\ =2\cos^{2}\left(\frac{nx}{4}\right)\sec\left(\frac{nx}{2}\right)$

###### Proof.

Use equation (5.1) and set $k=1,a=1,m=x,l=1,j=1$ and simplify using the method
in section (8.1) in [13]. ∎

######  Example 6.2.

A Finite Product Involving The Exponential Of Trigonometric Functions

(6.2) $\prod_{p=0}^{n-1}\left(\sin\left(\frac{\pi
p}{n}+\frac{x}{2}\right)\csc\left(\frac{\pi p}{n}+x\right)\right)^{i\pi}\\\
\exp\left(\cot\left(\frac{\pi p}{n}+\frac{x}{2}\right)-\cot\left(\frac{\pi
p}{n}+x\right)+2\sin\left(\frac{x}{2}\right)\right.\\\
\left.\left(2\cos\left(\frac{2\pi
p}{n}+\frac{3x}{2}\right)+i\pi\sin\left(\frac{2\pi
p}{n}+\frac{3x}{2}\right)\right)\right)\\\ \\\
=2^{-i\pi}\sec^{i\pi}\left(\frac{nx}{2}\right)(\sinh(n\csc(nx))+\cosh(n\csc(nx)))$

###### Proof.

Use equation (5.1) and set $k=1,a=-1,m=x,l=1,j=1$ and simplify using the
method in section (8.1) in [13]. ∎

######  Example 6.3.

A Finite Product Involving The Exponential Of Trigonometric Functions

(6.3) $\prod_{p=0}^{n-1}\sin^{3}\left(\frac{\pi
p}{n}+\frac{x}{2}\right)\csc^{2}\left(\frac{\pi
p}{n}+\frac{x}{4}\right)\csc\left(\frac{\pi p}{n}+x\right)\\\
\exp\left(-4\sin\left(\frac{x}{4}\right)\sin\left(\frac{2\pi
p}{n}+\frac{3x}{4}\right)-\frac{4}{3}\sin\left(\frac{3x}{4}\right)\right.\\\
\left.\sin\left(\frac{6\pi
p}{n}+\frac{9x}{4}\right)+\sin(x)\sin\left(\frac{4\pi
p}{n}+3x\right)+\frac{2}{3}\sin\left(\frac{3x}{2}\right)\right.\\\
\left.\sin\left(\frac{6\pi
p}{n}+\frac{9x}{2}\right)-4\sin\left(\frac{x}{2}\right)\sin\left(\frac{\pi
p}{n}\right)\cos\left(\frac{3\pi p}{n}+\frac{3x}{2}\right)\right)\\\ \\\
=\sec\left(\frac{nx}{2}\right)+1$

###### Proof.

Use equation (5.1) and set $k=1,a=1,m=x,l=1,j=3$ and simplify using the method
in section (8.1) in [13]. ∎

######  Example 6.4.

A Finite Product Involving The Exponential Of Trigonometric Functions

(6.4) $\prod_{p=0}^{n-1}\sin^{14}\left(\frac{2\pi
p}{n}+\frac{x}{4}\right)\sin\left(\frac{2\pi
p}{n}+x\right)\csc^{8}\left(\frac{2\pi
p}{n}+\frac{x}{8}\right)\csc^{7}\left(\frac{2\pi p}{n}+\frac{x}{2}\right)\\\
\exp\left(-8\cos\left(\frac{4\pi
p}{n}+\frac{x}{4}\right)+14\cos\left(\frac{4\pi
p}{n}+\frac{x}{2}\right)-7\cos\left(\frac{4\pi p}{n}+x\right)\right.\\\
\left.+\cos\left(\frac{4\pi p}{n}+2x\right)\right)\\\ \\\
=8\cos^{8}\left(\frac{nx}{8}\right)\cos\left(\frac{nx}{2}\right)\sec^{6}\left(\frac{nx}{4}\right)$

###### Proof.

Use equation (5.1) and set $k=2,a=1,m=x,l=2,j=1$ and simplify using the method
in section (8.1) in [13]. ∎

######  Example 6.5.

A Recurrence Identity With Consecutive Neighbours.

(6.5)
$\Phi(z,s,a)=z^{j}\left(z^{j+1}(-\Phi(z,s,a+2j+1))+2z\Phi(z,s,a+j+1)+i^{s}(i(a+j))^{-s}\right)$

###### Proof.

Use equation (5.1) and set $n=1,m=\log(z)/i,k=-s,a=e^{ai},j=0$ then set $a\to
2(a-1),z\to\sqrt{z}$ and simplify. ∎

######  Example 6.6.

A Recurrence Identity With Consecutive Neighbours.

(6.6) $\Phi(z,s,a)=z\Phi(z,s,a+1)+a^{-s}$

###### Proof.

Use equation (5.1) and set $n=1,m=\log(z)/i,k=-s,a=e^{ai},j=0$ and simplify.
This is equation (10.06.17.0002.01) in [14]. ∎

######  Example 6.7.

A Recurrence Identity With Consecutive Neighbours in terms of Glaisher’s
constant $A$.

(6.7)
$-\Phi^{\prime}(-i,-1,0)-\Phi^{\prime}(i,-1,0)-\text{Li}_{-1}^{\prime}(-i)-\text{Li}_{-1}^{\prime}(i)=\log\left(\frac{A^{24}}{16\
2^{2/3}e^{2}}\right)$

###### Proof.

Use equation (5.1) and set $n=2,a=1,j=0,m=\pi/4,l=3$ and simplify. Then take
the first partial derivative with respect to $k$ and set $k=1$ and simplify. ∎

######  Example 6.8.

A Finite Product Of Quotient Tangent Functions In Terms Of The Exponential Of
The Hurwitz-Lerch Zeta Function. Plots of the right-hand are produced for a
special case.

(6.8) $\prod_{p=0}^{n-1}\left(i\cot\left(\frac{\pi
lp+mn}{2n}\right)\right)^{2e^{-\frac{i(\pi
lp+mn)}{n}}}\left(-i\tan\left(\frac{\pi
lp+nr}{2n}\right)\right)^{2e^{-\frac{i(\pi lp+nr)}{n}}}\\\
=\exp\left(2e^{2imn}\Phi\left(e^{2imn},1,1+\frac{1}{2n}\right)-2e^{2inr}\Phi\left(e^{2inr},1,1+\frac{1}{2n}\right)\right)$

###### Proof.

Use equation (5.1) and form a second equation by replacing $m\to r$ and take
their difference. Next set $k=-1,a=e^{i},j=-1$ and simplify in terms of the
arctangent function using entry (3) in Table below (64:12:7) in [11]. Next
take the exponential of both sides and simplify. ∎

Figure 1.
$\text{Re}\left(\exp\left(2e^{2imn}\Phi\left(e^{2imn},1,1+\frac{1}{2n}\right)-2e^{2inr}\Phi\left(e^{2inr},1,1+\frac{1}{2n}\right)\right)\right)$
where $m,r\in\mathbb{R}$ and $n=5$ Figure 2.
$\text{Im}\left(\exp\left(2e^{2imn}\Phi\left(e^{2imn},1,1+\frac{1}{2n}\right)-2e^{2inr}\Phi\left(e^{2inr},1,1+\frac{1}{2n}\right)\right)\right)$.
where $m,r\in\mathbb{R}$ and $n=5$ Figure 3.
$\text{Abs}\left(\exp\left(2e^{2imn}\Phi\left(e^{2imn},1,1+\frac{1}{2n}\right)-2e^{2inr}\Phi\left(e^{2inr},1,1+\frac{1}{2n}\right)\right)\right)$.
where $m,r\in\mathbb{R}$ and $n=5$

######  Example 6.9.

Quotient Of Trigonometric Functions Raised To A Complex Power

(6.9)
$\prod_{p=0}^{n-1}\frac{\sin\left(\frac{lp\pi}{n}+\frac{x}{b}\right)}{\sin\left(\frac{lp\pi}{n}+x\right)}\left(\frac{\cos\left(\frac{lp\pi}{n}+\frac{x}{b}\right)\tan\left(\frac{lp\pi}{n}+\frac{x}{b}\right)}{\cos\left(\frac{lp\pi}{n}+\frac{x}{b^{2}}\right)\tan\left(\frac{lp\pi}{n}+\frac{x}{b^{2}}\right)}\right)^{b}\\\
=\frac{\sin\left(\frac{nx}{b}\right)}{\sin(nx)}\left(\frac{\cos\left(\frac{nx}{b}\right)\tan\left(\frac{nx}{b}\right)}{\cos\left(\frac{nx}{b^{2}}\right)\tan\left(\frac{nx}{b^{2}}\right)}\right)^{b}$

###### Proof.

Use equation (5.1) setting $k=1,a=1,m=x,j=-1$ and simplify using the method in
section (8.1) in [13]. ∎

######  Example 6.10.

Lerch Function Recurrence Identity With Consecutive Neighbours

(6.10) $e^{i\pi
jl}\left(\Phi(z,s,a-j)+z^{2j+1}\left(\Phi(z,s,a+j+1)+e^{i\pi(j+1)l}\Phi\left(e^{il\pi}z,s,a+j+1\right)\right)\right)\\\
+\Phi\left(e^{il\pi}z,s,a-j\right)\\\ =2e^{i\pi
jl}z^{j}\left(2^{1-s}z^{2}\Phi\left(z^{2},s,\frac{a}{2}+1\right)+e^{\frac{i\pi
s}{2}}(ia)^{-s}\right)$

###### Proof.

Use equation (5.1) and set $n=2,m=\log(z)/(2i),k=-s,a=e^{ai}$ and simplify. ∎

######  Example 6.11.

Special Case Of A Recurrence Lerch Function Identity

(6.11)
$\Phi^{\prime}(-i,0,a)+\Phi^{\prime}(i,0,a)=2\log\left(\frac{2\Gamma\left(\frac{a}{4}\right)}{(a-2)\Gamma\left(\frac{a-2}{4}\right)}\right)$

###### Proof.

Use equation (6.10) and set $a=2a,l=1,z=i,j=-1$ and simplify using equation
(25.14.2) in [3]. Next take the first partial derivative with respect to $s$
and set $s=0$ and simplify using equation (25.11.18 ) in [3]. ∎

######  Example 6.12.

Special Case Of A Recurrence Lerch Function Identity

(6.12)
$\Phi^{\prime}(-i,0,a)-\Phi^{\prime}(i,0,a)=i\log\left(\frac{4\Gamma\left(\frac{a+3}{4}\right)^{2}}{\Gamma\left(\frac{a+1}{4}\right)^{2}}\right)$

###### Proof.

Use equation (6.10) and set $a=2a,l=1,z=i,j=1$ and simplify using equation
(25.14.2) in [3]. Next take the first partial derivative with respect to $s$
and set $s=0$ and simplify using equation (25.11.18 ) in [3]. ∎

######  Example 6.13.

Special Case Of A Recurrence Lerch Function Identity

(6.13)
$\Phi^{\prime}(-i,0,a)^{2}-\Phi^{\prime}(i,0,a)^{2}=2i\log\left(\frac{\Gamma\left(\frac{a}{4}\right)}{2\Gamma\left(\frac{a+2}{4}\right)}\right)\log\left(\frac{4\Gamma\left(\frac{a+3}{4}\right)^{2}}{\Gamma\left(\frac{a+1}{4}\right)^{2}}\right)$

###### Proof.

Use equations (6.11) and (6.12) and multiply to yield quoted result. ∎

Figure 4.
$\text{Re}\left(2i\log\left(\frac{\Gamma\left(\frac{1}{4}(a+ib)\right)}{2\Gamma\left(\frac{1}{4}(a+ib+2)\right)}\right)\log\left(\frac{4\Gamma\left(\frac{1}{4}(a+ib+3)\right)^{2}}{\Gamma\left(\frac{1}{4}(a+ib+1)\right)^{2}}\right)\right)$
where $a,b\in\mathbb{R}$ Figure 5.
$\text{Im}\left(2i\log\left(\frac{\Gamma\left(\frac{1}{4}(a+ib)\right)}{2\Gamma\left(\frac{1}{4}(a+ib+2)\right)}\right)\log\left(\frac{4\Gamma\left(\frac{1}{4}(a+ib+3)\right)^{2}}{\Gamma\left(\frac{1}{4}(a+ib+1)\right)^{2}}\right)\right)$
where $a,b\in\mathbb{R}$ Figure 6.
$\text{Abs}\left(2i\log\left(\frac{\Gamma\left(\frac{1}{4}(a+ib)\right)}{2\Gamma\left(\frac{1}{4}(a+ib+2)\right)}\right)\log\left(\frac{4\Gamma\left(\frac{1}{4}(a+ib+3)\right)^{2}}{\Gamma\left(\frac{1}{4}(a+ib+1)\right)^{2}}\right)\right)$
where $a,b\in\mathbb{R}$

##  7\. Conclusion

In this paper, we have presented a method for deriving a finite sum of the
Hurwitz-Lerch Zeta function along with some interesting finite products of
trigonometric functions using contour integration. We will be applying this
method to other trigonometric formulae to derive other finite sums of the
Hurwitz-Lerch function in future work. The results presented were numerically
verified for both real and imaginary and complex values of the parameters in
the integrals using Mathematica by Wolfram.

## References

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http://functions.wolfram.com/10.06.17.0002.01


# The elliptical invariant tori of nearly integrable Hamiltonian system
through symplectic algorithms

Zaijiu Shang , Yang Xu

###### Abstract

In this paper we apply symplectic algorithms to nearly integrable Hamiltonian
system, and prove it can maintain lots of elliptic lower dimensional invariant
tori. We are committed to consider the elliptic lower dimensional invariant
tori for symplectic mapping with a small twist under the Rüssmann’s non-
degenerate condition, and focus on its measure estimation. And then apply it
to the nearly integrable Hamiltonian system to obtain lots of elliptic lower
dimensional invariant tori.

††Zaijiu Shang: Institute of Academy of Mathematics and Systems Science,
Chinese Academy of Sciences, Beijing 100080, China; e-mail<EMAIL_ADDRESS>
Yang Xu: School of Mathematical Sciences, Fudan University, Shanghai 200433,
China; e-mail<EMAIL_ADDRESS>
Keywords. nearly integrable, Symplectic algorithm, elliptic lower dimensional
invariant tori, Rüssmann’s non-degeneracy

###### Contents

  1. 1 Introduction
  2. 2 Preprocessing
  3. 3 Main Conclusions
  4. 4 The Symplectic Mapping System with a Small Twist ()
  5. 5 Measure estimation
  6. 6 Applied to the symplectic difference scheme (1.4)
    1. 6.1 Proof of Corollary 3.1
    2. 6.2 Proof of Corollary 3.2

##  1 Introduction

In section 1, we briefly describe three systems, they are the symplectic
mapping system with a small twist (1), the nearly integrable Hamiltonian
system (1.3) and its corresponding symplectic difference scheme (1.4). In
section 2, we give some notation. Section 3 contains the main conclusions. As
for the symplectic mapping system with a small twist (1), section 4 and
section 5 are proof of the existence of elliptic lower dimensional invariant
tori and measure estimation respectively. And applying this to the symplectic
difference scheme is covered in section 6.

Consider the symplectic mapping with a small twist

$F:(x,u,y,v)\in\mathbb{T}^{n}\times W\times V\times
W\to(\hat{x},\hat{u},\hat{y},\hat{v})\in\mathbb{T}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{n}\times\mathbb{R}^{m}$

where $\mathbb{T}^{n}$ is the usual $n$-torus, $V$ is a closed bounded and
connected domain, $V\subseteq\mathbb{R}^{n}$ is a domain around the origin.
The standard symplectic form is $\sum\limits_{i=1}^{n}dx_{i}\land
dy_{i}+\sum\limits_{j=1}^{m}du_{j}\land dv_{j}$. $F$ can be expressed
implicitly as

$\displaystyle\hat{x}$
$\displaystyle=x+\partial_{\hat{y}}tH(x,u,\hat{y},\hat{v})$ $\displaystyle y$
$\displaystyle=\hat{y}+\partial_{x}tH(x,u,\hat{y},\hat{v})$
$\displaystyle\hat{u}$
$\displaystyle=u+\partial_{\hat{v}}tH(x,u,\hat{y},\hat{v})$ $\displaystyle v$
$\displaystyle=\hat{v}+\partial_{u}tH(x,u,\hat{y},\hat{v})$ (1.1)

where $tH=tN+tP$, $t$ is the time step, and $tP$ is viewed as a perturbation.
The normal form $tN$ is expressed as

$tN(x,u,\hat{y},\hat{v})=th(\hat{y})+\langle
A_{t}u,\hat{v}\rangle+\frac{1}{2}\langle B_{t}u,u\rangle+\frac{1}{2}\langle
C_{t}\hat{v},\hat{v}\rangle$ (1.2)

For fixed $t$, $A_{t}=t(A-I),B_{t}=tB,C_{t}=tC$ are constant matrices, where
$I$ is the identity matrix of order $m$. Suppose $B$ and $C$ are symmetric.

More importantly, we can apply it to nearly integrable Hamiltonian system:

$F^{{}^{\prime}}:\dot{p}=\frac{\partial H_{\epsilon}}{\partial q}(p,q),\
\dot{q}=-\frac{\partial H_{\epsilon}}{\partial p}(p,q),\ (p,q)\in D$ (1.3)

where $H_{\epsilon}(p,q)=H_{0}(p,q)+\epsilon H_{1}(p,q,\epsilon)$ ,
$H_{0}(x,u,y,v)=h(y)+\langle Au,v\rangle+\frac{1}{2}\langle
Bu,u\rangle+\frac{1}{2}\langle Cv,v\rangle$, $p=(x,u)$, $q=(y,v)$.
$D\in{\mathbb{R}}^{2n}$ is a bounded connected open domain, the dot is the
derivative with respect to time $t$. Without loss of generality, we can assume
$\lvert H_{1}\rvert_{D}\leq M_{1}$, where $\lvert\cdot\lvert_{D}$ denotes the
supremum norm on $D$, $M_{1}$ is a constant, and we can ignore the parameter
expression of $\epsilon$.

Let us focus on the symplectic algorithm which is compatible with (1.3). By
Lemma 3.1 in [1], the system $G_{H_{\epsilon}}^{t}:(p,q)\to(\hat{p},\hat{q})$
can be expressed as follows:

$G_{H_{\epsilon}}^{t}:\begin{cases}\hat{p}=p+t\frac{\partial H_{0}}{\partial
q}(p,\hat{q})+t\epsilon\frac{\partial H_{1}}{\partial
q}(p,\hat{q})+t^{s+1}\frac{\partial{P_{1}}}{\partial q}(p,\hat{q})\\\
\hat{q}=q-t\frac{\partial
H_{0}}{\partial\hat{p}}(p,\hat{q})-t\epsilon\frac{\partial
H_{1}}{\partial\hat{p}}(p,\hat{q})-t^{s+1}\frac{\partial{P_{1}}}{\partial\hat{p}}(p,\hat{q})\end{cases}$
(1.4)

for $(p,\hat{q})\in D_{\frac{r}{4},\frac{s}{4}}$, where $s$ is a positive
constant. Here $\delta$ is a sufficiently small positive number such that
$P_{1}$ , which depends on time step $t$, is well defined and real analytic on
$D_{\frac{r}{4},\frac{s}{4}}$ for $t\in[0,\delta]$, where $D_{r,s}$ is a
complex neighborhood of $D$ with $D_{r,s}=\\{p:\lvert
p\rvert<r\\}\times\\{q:\lvert\Im
q\rvert<s\\}\subset\mathbb{C}^{n}\times\mathbb{C}^{n}$. By Lemma 3.1 and Lemma
3.3 in [1], $\big{\lvert}\frac{\partial
P_{1}}{\partial{\hat{p}}}\big{\rvert}_{\frac{r}{4},\frac{s}{4}}\leq M_{2}$,
$\big{\lvert}\frac{\partial P_{1}}{\partial
q}\big{\rvert}_{\frac{r}{4},\frac{s}{4}}\leq M_{2}$, where
$\lvert\cdot\lvert_{\frac{r}{4},\frac{s}{4}}$ denotes the supremum norm on
$D_{\frac{r}{4},\frac{s}{4}}$, $M_{2}$ is a constant. And similar to [1], fix
$(p_{0},\hat{q_{0}})\in D$, let $P_{1}(p_{0},\hat{q_{0}})=0$, for
$(p,\hat{q})\in D_{\frac{r}{4},\frac{s}{4}}$, we have $\lvert
P_{1}\rvert_{\frac{r}{4},\frac{s}{4}}\leq 2nM_{2}l_{*}$ for $t\in[0,\delta]$,
where $l_{*}$ is an upper bound of the length of the shortest curves from
$(p_{0},\hat{q_{0}})$ to $(p,\hat{q})$ in $D_{\frac{r}{4},\frac{s}{4}}$.

By the way, for the nearly integrable Hamiltonian system (1.3), there exists a
generating function $\widetilde{S}$ such that
$H_{1}(p,q)-\widetilde{S}(p,q)\sim o(t^{s})$.

First we research the symplectic mapping system with a small twist (1), find
its elliptic lower dimensional invariant tori, and give measure estimations.
And then we applied it to the nearly integrable Hamiltonian system (1.3). For
the symplectic difference scheme (1.4) which is compatible with (1.3), $S_{0}$
is regarded as $N$, $\epsilon H_{1}+t^{s}P_{1}$ as $P$, and the conclusion of
system (1) is applied. So we pay more attention to the symplectic mapping
system with a small twist (1).

##  2 Preprocessing

As for the symplectic mapping system with a small twist (1), if $P=0$, then

$\displaystyle\hat{x}$ $\displaystyle=x+t\omega(\hat{y})$ $\displaystyle y$
$\displaystyle=\hat{y}$ $\displaystyle\hat{u}$
$\displaystyle=\underline{A_{t}}u+C_{t}\hat{v}$ $\displaystyle v$
$\displaystyle=\underline{A_{t}}^{T}\hat{v}+B_{t}u$

where $\omega(\hat{y})=h^{{}^{\prime}}(\hat{y})$,
$\underline{A_{t}}=I+A_{t}=I+t(A-I)$, $\underline{A_{t}}^{T}$ is the transpose
of $\underline{A_{t}}$.

For $\underline{A_{t}}$ is non-singular, $F$ can be expressed as

$\displaystyle\hat{x}$ $\displaystyle=x+t\omega(y)$ $\displaystyle\hat{y}$
$\displaystyle=y$ $\displaystyle\hat{u}$
$\displaystyle=\big{(}\underline{A_{t}}-C_{t}(\underline{A_{t}}^{T})^{-1}B_{t}\big{)}u+C_{t}(\underline{A_{t}}^{T})^{-1}v$
$\displaystyle\hat{v}$
$\displaystyle=-(\underline{A_{t}}^{T})^{-1}B_{t}u+(\underline{A_{t}}^{T})^{-1}v$

Let

$\Omega=\left(\begin{array}[]{cc}\underline{A_{t}}-C_{t}(\underline{A_{t}}^{T})^{-1}B_{t}&\
C_{t}(\underline{A_{t}}^{T})^{-1}\\\
-(\underline{A_{t}}^{T})^{-1}B_{t}&(\underline{A_{t}}^{T})^{-1}\end{array}\right)_{2m\times
2m}$

For $\xi\in V$, it admits a m-dimensional invariant torus
$\mathbb{T}_{\xi}=\mathbb{T}^{n}\times\\{0\\}\times\\{\xi\\}\times\\{0\\}$,
$t\omega(\xi)$ is the rotational frequency.

In this paper, we consider the small perturbation, i.e.$P\neq 0$.

Set

$\underline{A_{t}}=\sec{\theta^{t}}\ \ \ B_{t}=C_{t}=\tan{\theta^{t}}$ (2.1)

where $\theta^{t}=(\theta_{1}^{t},\theta_{2}^{t},\dots,\theta_{m}^{t})$ and
$\sec{\theta^{t}}=diag(\sec{\theta_{1}^{t}},\sec{\theta_{2}^{t}},\dots,\sec{\theta_{m}^{t}})$.
Then,

$\Omega=\left(\begin{array}[]{cc}\cos{\theta^{t}}&\ \sin{\theta^{t}}\\\
-\sin{\theta^{t}}&\ \cos{\theta^{t}}\end{array}\right)_{2m\times 2m}$

It is easy to see that $\Omega$ has the eigenvalues $e^{\pm
i\theta_{1}^{t}},e^{\pm i\theta_{2}^{t}},\dots,e^{\pm i\theta_{m}^{t}}$. Let
us assume that the following non-resonant conditions hold for
$l,i,j\in\mathbb{Z}$ and $1\leq i,j\leq m$.

$\begin{cases}\theta_{j}^{t}-2\pi l\neq 0\\\
\theta_{j}^{t}\pm\theta_{i}^{t}-2\pi l\neq 0,\ |l|+|i-j|\neq 0\end{cases}$
(2.2)

In this paper, we consider $h(\xi)$ is analytic in $\xi\in V+r$, where
$V+r=\bigcup\limits_{b\in
V}\\{\xi\in\mathbb{C}^{n}:\lvert\xi-b\rvert_{2}<r\\}\subseteq\mathbb{C}^{n}$,
and $\omega(\xi)=h^{{}^{\prime}}(\xi)$ satisfies the Rüssmann’s non-degeneracy
condition and satisfies

$\lvert\omega(\xi_{1})-\omega(\xi_{2})\rvert\leq\Theta\lvert\xi_{1}-\xi_{2}\rvert,\
\xi_{1},\xi_{2}\in V+r.$ (2.3)

######  Remark 2.1.

For $\omega$ satisfies the Rüssmann’s non-degeneracy condition, that is
$\langle\omega,y\rangle\not\equiv 0$ for all
$y\in\mathbb{R}^{n}\setminus\\{0\\}$. In analytic cases, [2] gave an
expression of the Rüssmann’s non-degeneracy condition (see Remark 3.1 in [2]),
that is, if the Rüssmann’s non-degeneracy is satisfied, then $\exists\
\bar{n}\in\mathbb{N}$ such that

$rank\big{\\{}\partial^{i}_{\xi}\omega(\xi):|i|\leq\bar{n}\big{\\}}=n,\
\forall\ \xi\in V$ (2.4)

And, by Rüssmann (see Lemma 18.2 in [3]), there exist
$\bar{n}(V)\in\mathbb{N},\beta_{0}(V)>0$ such that

$\min\limits_{\xi\in V}\max_{0\leq v\leq\bar{n}}\lvert D^{v}\langle
k,\omega(\xi)\rangle\rvert\geq\beta_{0}$ (2.5)

where $\langle k,\omega\rangle=\sum\limits_{j=1}^{n}k_{j}\omega_{j}$, $\lvert
k_{j}\rvert=1,j=1,\dots,n$. And we can take the smallest of such integer
$\bar{n}$. And by Rüssmann (Theorem 18.4 in [3]), we have

$\min_{\xi\in V}\max_{0\leq v\leq\bar{n}}\big{\lvert}D^{v}{\lvert
k\rvert_{2}}^{-2}|\langle k,\omega(\xi)\rangle|^{2}\big{\rvert}\geq\beta_{0}$
(2.6)

for $\forall k\in\mathbb{Z}^{n}\setminus\\{0\\}$. So, by $\lvert\cdot\rvert$
and $\lvert\cdot\rvert_{2}$ are equivalent, where $\lvert
k\rvert=\sum\limits_{j=1}^{n}\lvert k_{j}\rvert$, $\lvert
k\rvert_{2}=(\sum\limits_{j=1}^{n}\lvert k_{j}\rvert^{2})^{\frac{1}{2}}$,
there exist $\bar{n}$, $\beta$ such that

$\min_{\xi\in V}\max_{0\leq v\leq\bar{n}}|D^{v}\langle
k,\omega(\xi)\rangle|\geq\beta\lvert k\rvert$ (2.7)

for $\forall k\in\mathbb{Z}^{n}\setminus\\{0\\}$, here
$\bar{n}=\bar{n}(\omega,V)\in\mathbb{N}$ and $\beta=\beta(\omega,V)>0$ are
called the index and amount of $\omega$ with respect to $V$ respectively.

Now, we give some notations. Set

$T_{s}=\\{x\in\mathbb{C}^{n}/2\pi\mathbb{Z}^{n}:|\Im x|_{\infty}\leq s\\}$
$V_{r}=\\{y\in\mathbb{C}^{n}:|y|_{1}\leq r^{2}\\}\ \
W_{r}=\\{w\in\mathbb{C}^{m}:|w|_{2}\leq r\\}$

Consider $D(s,r)=T_{s}\times W_{r}\times V_{r}\times W_{r}$. Here,
$|x|_{\infty}=\max\limits_{1\leq j\leq
n}|x_{j}|,|y|_{1}=\sum\limits_{j=1}^{n}|y_{j}|$, and
$|w|_{2}=(\sum\limits_{j=1}^{m}|w_{j}|^{2})^{\frac{1}{2}}$.

Expand $P(x;\xi)$ into Fourier series
$P(x;\xi)=\sum\limits_{k\in\mathbb{Z}^{n}}P_{k}(\xi)e^{i\langle k,x\rangle}$,
supposing $P(x,u,\hat{y},\hat{v};\xi)$ is analytic on $D(s,r)$ with respect to
$(x,u,\hat{y},\hat{v})$. Define

$\lVert P\rVert^{*}_{s}=\sum\limits_{k\in\mathbb{Z}^{n}}\lVert
P_{k}\rVert^{*}_{V}e^{s|k|},with\ \lVert
P_{k}\rVert^{*}_{V}=\max\limits_{|i|\leq\bar{n}}\max\limits_{\xi\in
V}|\partial^{i}_{\xi}P_{k}(\xi)|$ $\lVert
P\rVert^{*}_{s,r}=\sum\limits_{k\in\mathbb{Z}^{n}}|P_{k}|^{*}_{r}e^{s|k|},with\
|P_{k}|^{*}_{r}=\sup\limits_{(u,\hat{y},\hat{v})\in W_{r}\times V_{r}\times
W_{r}}\sum\limits_{i,j,l}\lVert
P_{klij}\rVert^{*}_{s}{\hat{y}}^{l}u^{i}{\hat{v}}^{j}$

Define
$X_{p}=(-\partial_{\hat{y}}P,-\partial_{\hat{v}}P,\partial_{x}P,\partial_{u}P)$.
Define the weighed norm by

$|\lVert
X_{P}\rVert|_{r;D(s,r)}^{*}=\lVert\partial_{\hat{y}}P\rVert_{s,r}^{*}+\frac{1}{r}\lVert\partial_{\hat{v}}P\rVert_{s,r}^{*}+\frac{1}{r^{2}}\lVert\partial_{x}P\rVert_{s,r}^{*}+\frac{1}{r}\lVert\partial_{u}P\rVert_{s,r}^{*}$

Where
$\lVert\partial_{\hat{x}}P\rVert_{s,r}^{*}=\sum\limits_{j}\lVert\partial_{\hat{x}_{j}}P\rVert_{s,r}^{*}$,
$\lVert\partial_{\hat{y}}P\rVert_{s,r}^{*}=\max\limits_{j}\lVert\partial_{\hat{y}_{j}}P\rVert_{s,r}^{*}$,
and
$\lVert\partial_{u}P\rVert_{s,r}^{*}=(\sum\limits_{j}\lVert\partial_{u_{j}}P\rVert_{s,r}^{*})^{\frac{1}{2}}$.
$\lVert\partial_{\hat{v}}P\rVert_{s,r}^{*}$ is similar to
$\lVert\partial_{u}P\rVert_{s,r}^{*}$.

Fix constant $\tau$ such that $\tau\geq(n+2)(\bar{n}+1)$ and
$\tau\geq(n+2)L+1$, where $L$ is a constant in 5 Measure Estimation.

##  3 Main Conclusions

For the symplectic mapping system with a small twist (1), we have two main
theorems.

######  Theorem 3.1.

For the symplectic mapping $F$ defined by (1) , $tH=tN+tP$, assume $h$ and $P$
are real analytic on a complex neighborhood $D$ of $\mathbb{T}^{n}\times
W\times V\times W$, $\omega$ satisfies the Rüssmann’s non-degeneracy condition
and (2.3), and (2.2) holds, $\sup_{D}|P(x,u,\hat{y},\hat{v})|\leq\epsilon$ for
$\epsilon>0$ small enough, and time step $t$ is small enough. Then, there is a
Cantor-like subset $V_{\gamma}\subseteq V$ non-empty such that $F$ maintains a
cluster of invariant torus $\\{\mathbb{T}_{\xi}\\}_{\xi\in V_{\gamma}}$.
Moreover, $meas(V\setminus V_{\gamma})\to 0$ as $\gamma\to 0$.

######  Theorem 3.2.

For the symplectic mapping $F$ defined by (1), and for the previous
$tN,A_{t},B_{t},C_{t}$, suppose $\omega$ satisfies Rüssmann’s non-degeneracy
condition and (2.3), $A_{t},B_{t},C_{t}$ satisfy the non-resonant condition
(2.2), $P$ is real analytic on $D(s,r)\times V$ with $\xi\in V$, $t$ is small
enough. Then, for $\gamma>0$, $\exists\ \epsilon_{0}>0$ small enough , such
that, if $|\lVert X_{P}\rVert|_{r;D(s,r)}^{*}=\epsilon\leq\epsilon_{0}$, then,
$\exists\ V_{\gamma}\subseteq V$ Cantor-like and non-empty, and there is a
symplectic mapping $\Psi$, such that $F_{\infty}={\Psi}^{-1}\circ F\circ\Psi$
which is generated by $tH_{*}=tN_{*}+tP_{*}$, where

$\displaystyle tN_{*}(x,u,\hat{y},\hat{v};\xi)$ $\displaystyle=\langle
t\omega_{\infty},\hat{y}\rangle+\langle
A_{t}^{\infty}u,\hat{v}\rangle+\frac{1}{2}\langle
B_{t}^{\infty}u,u\rangle+\frac{1}{2}\langle
C_{t}^{\infty}\hat{v},\hat{v}\rangle$ $\displaystyle
tP_{*}(x,u,\hat{y},\hat{v};\xi)$ $\displaystyle=t\sum\limits_{|i|+|j|+2|l|\geq
3}P_{lij}(x;\xi){\hat{y}}^{l}u^{i}{\hat{v}}^{j}$

with $\omega_{\infty}-\omega=O(\epsilon)$, $A_{t}^{\infty}-A_{t}=O(\epsilon)$,
$B_{t}^{\infty}-B_{t}=O(\epsilon)$, $\Psi-id=O(\epsilon)$. Then for each
$\xi\in V_{\gamma}$, there exists an invariant torus $\mathbb{T}_{\xi}$.
Moreover, $meas(V\setminus V_{\gamma})\to 0$ as $\gamma\to 0$. To be specific,

$\lvert V\setminus V_{\gamma}\rvert\leq c_{3}\gamma
d^{n}((n+1)^{-\frac{1}{2}}+2d+\theta^{-1}d)+c_{5}\gamma d^{n}$

where $c_{3}=12\bar{c}(2\pi
e)^{\frac{n+1}{2}}(\bar{n}+2)^{\bar{n}+3}[(\bar{n}+2)!]^{-1}\big{(}\lvert\omega\rvert_{\mathcal{K}}+c\epsilon+1\big{)}\beta^{-\frac{\bar{n}+2}{\bar{n}+1}}\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}$,
$c_{5}=4\bar{\bar{c}}c_{4}m(\lvert\omega\rvert_{V}+c\epsilon+\max\limits_{1\leq
j\leq m}3|B^{0}_{j}|+1)$, $\mathcal{B}=(V\times(0,1)+r)\cap\mathbb{R}^{n+1}$,
$d$ is the diameter of $V\times(0,1)$, $\bar{c}$ and $\bar{\bar{c}}$ depend
only on $n$, $\bar{n}$ and $\tau$, $c_{4}$ and $c$ are independent of $t$,
$\epsilon$ and $\gamma$.

Applying these conclusion to the symplectic difference scheme (1.4), we get
two main corollaries.

######  Corollary 3.1.

For the nearly integrable Hamiltonian system (1.3) and its corresponding
symplectic difference scheme (1.4) , $H_{\epsilon}=H_{0}+\epsilon H_{1}$,
$tH_{0}=tN^{{}^{\prime}}$ with the form (1.2), assume $h$ is real analytic on
a complex neighborhood $D$ of $\mathbb{T}^{n}\times W\times V\times W$,
$\omega$ satisfies the Rüssmann’s non-degeneracy condition and (2.3),
$A_{t},B_{t},C_{t}$ satisfy the non-resonant condition (2.2), $P$ is real
analytic on $D(s,r)\times V$ with $\xi\in V$, $\sup_{D}|\epsilon
H_{1}+t^{s}P_{1}|\leq\epsilon^{{}^{\prime}}$ for $\epsilon^{{}^{\prime}}>0$.
We can denote $\epsilon H_{1}+t^{s}P_{1}=P^{{}^{\prime}}$. Then, for
$\epsilon$ and time step $t$ small enough, most invariant tori of the nearly
integrable system do not vanish, but are only slightly deformed, such that the
symplectic difference scheme also has invariant tori. There exist a Cantor
subset $V_{\epsilon,t}\subseteq V$, a mapping
$\omega_{\epsilon,t}:V_{\epsilon,t}\to\Omega_{\epsilon,t}$, and a symplectic
mapping
$\Psi_{\epsilon,t}:V_{\epsilon,t}\times\mathbb{T}^{n}\to\mathbb{R}^{n}\times\mathbb{T}^{n}$,
in the sense of Whitney, such that

  1. (i)

$G_{\infty}=\Psi_{\epsilon,t}^{-1}\circ
G_{H_{\epsilon}}^{t}\circ\Psi_{\epsilon,t}$ which is generated by
$t(H_{\epsilon})_{*}=tN^{{}^{\prime}}_{*}+tP^{{}^{\prime}}_{*}$, where

$\displaystyle tN^{{}^{\prime}}_{*}(x,u,\hat{y},\hat{v};\xi)$
$\displaystyle=\langle t\omega_{\epsilon,t},\hat{y}\rangle+\langle
A_{t}^{\infty}u,\hat{v}\rangle+\frac{1}{2}\langle
B_{t}^{\infty}u,u\rangle+\frac{1}{2}\langle
C_{t}^{\infty}\hat{v},\hat{v}\rangle$ $\displaystyle
tP^{{}^{\prime}}_{*}(x,u,\hat{y},\hat{v};\xi)$
$\displaystyle=t\sum\limits_{|i|+|j|+2|l|\geq
3}P^{{}^{\prime}}_{lij}(x;\xi){\hat{y}}^{l}u^{i}{\hat{v}}^{j}$

with $\omega_{\epsilon,t}-\omega=O(\epsilon^{{}^{\prime}})$,
$A_{t}^{\infty}-A_{t}=O(\epsilon^{{}^{\prime}})$,
$B_{t}^{\infty}-B_{t}=O(\epsilon^{{}^{\prime}})$,$\Psi_{\epsilon,t}-id=O(\epsilon^{{}^{\prime}})$.
Then for each $\xi\in V_{\epsilon,t}$, there exists an invariant torus
$\mathbb{T}_{\xi}$.

  2. (ii)

$meas(V\setminus V_{\epsilon,t})\to 0$ as $\gamma\to 0$. To be specific,
$V_{\epsilon,t}$ is a set of positive measure with the measure estimate

$\lvert V\setminus V_{\epsilon,t}\rvert\leq c_{3}^{{}^{\prime}}\gamma
d^{n}((n+1)^{-\frac{1}{2}}+2d+\theta^{-1}d)+c_{5}^{{}^{\prime}}\gamma d^{n}$
(3.1)

where $c_{3}^{{}^{\prime}}=12\bar{c}(2\pi
e)^{\frac{n+1}{2}}(\bar{n}+2)^{\bar{n}+3}[(\bar{n}+2)!]^{-1}\big{(}\lvert\omega\rvert_{\mathcal{K}}+c\epsilon^{{}^{\prime}}+1\big{)}\beta^{-\frac{\bar{n}+2}{\bar{n}+1}}\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}$,
$c_{5}^{{}^{\prime}}=4\bar{\bar{c}}c_{4}m(\lvert\omega\rvert_{V}+c\epsilon^{{}^{\prime}}+\max\limits_{1\leq
j\leq m}3|B^{0}_{j}|+1)$, $\mathcal{B}$, $d$, $\bar{c}$, $\bar{\bar{c}}$ ,
$c_{4}$ and $c$ are in Theorem 3.2.

######  Corollary 3.2.

Let the time step be $t_{1}$ and $t_{2}$ respectively, and comppare the two
systems, then for $\xi\in V_{\epsilon,t_{1}}\cap V_{\epsilon,t_{2}}$ we have

$\displaystyle\lvert\Psi_{\epsilon,t_{1}}-\Psi_{\epsilon,t_{2}}\rvert\leq
c(t_{1}^{\frac{s}{4}}-t_{2}^{\frac{s}{4}}),\
\lvert\omega_{\epsilon,t_{1}}-\omega_{\epsilon,t_{2}}\rvert\leq
c(t_{1}^{\frac{s}{2}}-t_{2}^{\frac{s}{2}}).$

$V_{\gamma,t_{1}}\cap V_{\gamma,t_{2}}$ is also a set of positive measure if
$\gamma$ is small enough. And for $\gamma\to 0$, we also have $\lvert
V\setminus(V_{\gamma,t_{1}}\cap V_{\gamma,t_{2}})\rvert\to 0$.

##  4 The Symplectic Mapping System with a Small Twist (1)

Let us focus on the symplectic mapping system with a small twist (1). The
proof of Theorem 3.1 and Theorem 3.2 are similar to the proof of Theorem 1.1
and Theorem 1.2 in [5], so we will not write them in detail here. we are
mainly concerned with measure estimation.

The symplectic mapping in [5] is expressed as

$\displaystyle\hat{x}$
$\displaystyle=x+\partial_{\hat{y}}H(x,u,\hat{y},\hat{v})$ $\displaystyle y$
$\displaystyle=\hat{y}+\partial_{x}H(x,u,\hat{y},\hat{v})$
$\displaystyle\hat{u}$
$\displaystyle=u+\partial_{\hat{v}}H(x,u,\hat{y},\hat{v})$ $\displaystyle v$
$\displaystyle=\hat{v}+\partial_{u}H(x,u,\hat{y},\hat{v})$

where $H=N+P$, and

$N(x,u,\hat{y},\hat{v})=h(\hat{y})+\langle(A-I)u,\hat{v}\rangle+\frac{1}{2}\langle
Bu,u\rangle+\frac{1}{2}\langle C\hat{v},\hat{v}\rangle$

We introduced a positive step $t$, and we want to explore the system as $t\to
0$.

Now, we briefly describe the proving process of the Theorem 3.1 and Theorem
3.2, please refer to [5] for details. Write $x_{+}$ as $x_{v+1}$ for
simplicity, and everything else is similar.

Let $p=(x,u),q=(y,v)$. We have $F:(p,q)\to(\hat{p},\hat{q})$

$\hat{p}=p+\partial_{\hat{q}}tH(p,\hat{q})\ \ \
q=\hat{q}+\partial_{p}tH(p,\hat{q})$ (4.1)

Suppose that $\Psi$ is close to the identity,
$\Psi:(p_{+},q_{+})\to(\hat{p},\hat{q})$

$q=q_{+}+F_{1}(p,q_{+})\ \ \ p_{+}=p+F_{2}(p,q_{+})$ (4.2)

then we have $F_{+}={\Psi}^{-1}\circ
F\circ\Psi:(p_{+},q_{+})\to(\hat{p}_{+},\hat{q}_{+})$, and it can be expressed
implicitly by

$\hat{p}_{+}=p+tH_{2}(p,\hat{q})+F_{2}(\hat{p},\hat{q}_{+})\ \ \
q_{+}=\hat{q}+tH_{1}(p,\hat{q})-F_{1}(p,q_{+})$ (4.3)

Then, by a similar but slightly different processing from Lemma 2.1 in [5],
$F_{+}$ can be expressed implicitly as

$\hat{p}_{+}=p_{+}+\partial_{\hat{q}_{+}}tH_{+}(p_{+},\hat{q}_{+})\ \ \
q_{+}=\hat{q}_{+}+\partial_{p_{+}}tH_{+}(p_{+},\hat{q}_{+})$ (4.4)

where $tH_{+}$ has the following form

$\displaystyle tH_{+}(p_{+},\hat{q}_{+})=$ $\displaystyle\
tH(p,\hat{q})+tH_{1}(p,\hat{q})F_{2}(p,q_{+})-tH_{2}(p,\hat{q})F_{1}(\hat{p},\hat{q}_{+})+F(\hat{p},\hat{q}_{+})$
$\displaystyle-F(p,q_{+})-F_{1}(p,q_{+})F_{2}(p,q_{+})+F_{1}(\hat{p},\hat{q}_{+})F_{2}(p,q_{+})$

And for $tH=tN+tP$, by a similar analysis to [5], let $z=(p_{+},\hat{q}_{+})$
then we can get

$tH_{+}(z)=tH(z)+F(p_{+}+tN_{2}(z),\hat{q}_{+})-F(p_{+},\hat{q}_{+}+tN_{1}(z))+Q(z)$
(4.5)

where $Q$ is a higher order small quantity.

######  Remark 4.1.

To be specific, by some similar but slightly different analysis from [5], we
have these following expressions. From the foregoing we have let
$z=(p_{+},\hat{q}_{+})$, here we let $\Delta
z=(-F_{2}(p,q_{+}),F_{1}(\hat{p},\hat{q}_{+}))$, then $(p,\hat{q})=z+\Delta
z$. So,

$tH(p,\hat{q})=tH(z+\Delta z)=tH(z)+\langle\nabla tH(z),\Delta z\rangle-Q_{1}$

where

$Q_{1}=-\frac{1}{2}\int_{0}^{1}\langle D^{2}tH(z_{s})\Delta z,\Delta z\rangle
ds,\ z_{s}=z+s\Delta z$

here $\nabla tH$ means the gradient of $tH$, and $D^{2}tH$ indicates the
Hessian matrix. By

$\nabla tH(z+\Delta z)=\nabla tH(z)+\int_{0}^{1}D^{2}tH(z_{s})\Delta zds$

we have

$\displaystyle
tH_{1}(p,\hat{q})F_{2}(p,q_{+})-tH_{2}(p,\hat{q})F_{1}(\hat{p},\hat{q}_{+})$
$\displaystyle=$ $\displaystyle-\langle\nabla
tH(p,\hat{q}),(-F_{2}(p,q_{+}),F_{1}(\hat{p},\hat{q}_{+}))\rangle$
$\displaystyle=$ $\displaystyle-\langle\nabla tH(z+\Delta z),\Delta
z\rangle=-\langle\nabla tH(z),\Delta z\rangle+2Q_{1}$

Note that

$\displaystyle\hat{p}=p_{+}+tN_{2}(z+\Delta
z)+tP_{2}(p,\hat{q})-F_{2}(p,q_{+})$ $\displaystyle tN_{2}(z+\Delta
z)=tN_{2}(z)+\int_{0}^{1}DtN_{2}(z_{s})\Delta zds$

then we have

$\displaystyle F(p_{+}+tN_{2}(z),\hat{q}_{+})-F(\hat{p},\hat{q}_{+})$
$\displaystyle=$ $\displaystyle\
Q_{2}+F_{1}(\hat{p},\hat{q}_{+})F_{2}(p,q_{+})+\sum\limits_{i=2}^{\infty}\frac{1}{i!}F_{1^{i}}(\hat{p},\hat{q}_{+})\cdot\big{(}F_{2}(p,q_{+})\big{)}^{i}$

where

$Q_{2}=\int_{0}^{1}F_{1}(z+s\Delta w)\Delta wds,\ \Delta
w=tP_{2}(p,\hat{q})+\int_{0}^{1}DtN_{2}(z_{s})\Delta zds$

where $DtN_{2}$ indicates the Jacobian matrix, and
$F_{1^{i}}(\hat{p},\hat{q}_{+})$ means the $i$-th partial of
$F(\hat{p},\hat{q}_{+})$ with respect to $\hat{p}$. Similarly, by
$p_{+}=p+F_{2}(p,q_{+})$, we have

$\displaystyle F(p_{+},q_{+})-F(p,q_{+})$
$\displaystyle=F_{1}(p,q_{+})F_{2}(p,q_{+})+\sum\limits_{i=2}^{\infty}\frac{1}{i!}F_{1^{i}}(p,q_{+})\big{(}F_{2}(p,q_{+})\big{)}^{i}$
$\displaystyle F(p_{+},\hat{q}_{+}+tN_{1}(z))-F(p_{+},q_{+})=Q_{3}$

where

$Q_{3}=\int_{0}^{1}F_{2}(z+s\Delta\omega^{{}^{\prime}})\Delta\omega^{{}^{\prime}}ds,\
\Delta\omega^{{}^{\prime}}=\partial_{1}\big{(}tH_{+}(z)-tH(z)+tP(z)\big{)}$

And we have

$\sum\limits_{i=2}^{\infty}\frac{1}{i!}F_{1^{i}}(\hat{p},\hat{q}_{+})\cdot\big{(}F_{2}(p,q_{+})\big{)}^{i}-\sum\limits_{i=2}^{\infty}\frac{1}{i!}F_{1^{i}}(p,q_{+})\big{(}F_{2}(p,q_{+})\big{)}^{i}=Q_{4}$

where

$Q_{4}=\big{(}F(\hat{p},\hat{q}_{+})-F(p,q_{+})\big{)}-\big{(}F_{1}(\hat{p},\hat{q}_{+})-F_{1}(p,q_{+})\big{)}F_{2}(p,q_{+})$

So, $Q=Q_{1}-Q_{2}+Q_{3}-Q_{4}$.

Next, similar to [5], we truncate $P$ with $P=R+\widetilde{P}$, where

$\displaystyle R(p,\hat{q})=P_{000}(x)+\langle
P_{100}(x),\hat{y}\rangle+\langle P_{010}(x),u\rangle+\langle
P_{001}(x),\hat{v}\rangle$ $\displaystyle+\langle
P_{011}(x)u,\hat{v}\rangle+\frac{1}{2}\langle
P_{020}(x)u,u\rangle+\frac{1}{2}\langle P_{002}(x)\hat{v},\hat{v}\rangle$
(4.6)

with the coefficients
$P_{lij}=\frac{\partial^{l+i+j}P}{\partial{{\hat{y}^{l}}\partial{u^{i}}\partial{{\hat{v}^{j}}}}}\big{|}_{u=0,\hat{y}=0,\hat{v}=0}$,
for $2l+i+j\leq 2$. Then (4.5) leads to the following equation:

$\displaystyle tR(p_{+},\hat{q}_{+})-$ $\displaystyle
F(p_{+},\hat{q}_{+}+tN_{p}(p_{+},\hat{q}_{+}))+F(p_{+}+tN_{q}(p_{+},\hat{q}_{+}),\hat{q}_{+})$
$\displaystyle+tN(p_{+},\hat{q}_{+})=t\bar{N}(p_{+},\hat{q}_{+})$

Where $t\bar{N}(p_{+},\hat{q}_{+})$ represent the new normal form.

Then, similar to [5], let us use a linear operator, which is defined as
$LF=F(p+tN_{\hat{q}}(p,\hat{q}),\hat{q})-F(p,\hat{q}+tN_{p}(p,\hat{q}))$. Let
$F(p,\hat{q})$ remain in the form of $R(p,\hat{q})$ in (4), then we have

$\displaystyle F(p,\hat{q}+tN_{p}(p,\hat{q}))$
$\displaystyle=F_{000}(x)+\langle F_{100}(x),\hat{y}\rangle+\langle
F_{010}(x),u\rangle+\langle B_{t}F_{001}(x),u\rangle$
$\displaystyle+\langle\underline{A_{t}}F_{001}(x),\hat{v}\rangle+\langle\underline{A_{t}}F_{011}(x)u,\hat{v}\rangle+\langle\underline{A_{t}}F_{002}(x)B_{t}u,\hat{v}\rangle$
$\displaystyle+\langle B_{t}F_{011}(x)u,u\rangle+\frac{1}{2}\langle
B_{t}F_{002}(x)B_{t}u,u\rangle$
$\displaystyle+\frac{1}{2}\langle\underline{A_{t}}F_{002}(x){\underline{A_{t}}}^{T}\hat{v},\hat{v}\rangle+\frac{1}{2}\langle
F_{020}(x)u,u\rangle$ $\displaystyle F(p+tN_{\hat{q}}(p,\hat{q}),\hat{q})$
$\displaystyle=F_{000}(\tilde{x})+\langle
F_{100}(\tilde{x}),\hat{y}\rangle+\langle
F_{010}(\tilde{x}),\hat{v}\rangle+\langle
C_{t}F_{001}(\tilde{x}),\hat{v}\rangle$
$\displaystyle+\langle{\underline{A_{t}}}^{T}F_{010}(\tilde{x}),u\rangle+\langle
F_{011}(\tilde{x})\underline{A_{t}}u,\hat{v}\rangle+\langle
C_{t}F_{020}(\tilde{x})\underline{A_{t}}u,\hat{v}\rangle$
$\displaystyle+\langle
F_{011}(\tilde{x})C_{t}\hat{v},\hat{v}\rangle+\frac{1}{2}\langle
C_{t}F_{020}(\tilde{x})C_{t}\hat{v},\hat{v}\rangle$
$\displaystyle+\frac{1}{2}\langle{\underline{A_{t}}}^{T}F_{020}(\tilde{x})\underline{A_{t}}u,u\rangle+\frac{1}{2}\langle
F_{002}(\tilde{x})\hat{v},\hat{v}\rangle$

where $\tilde{x}=x+t\omega$ , $R_{020},R_{002},F_{020}$ and $F_{002}$ are
symmetric. So

$LF=L_{0}+\langle L_{1},\hat{y}\rangle+\langle L_{10},u\rangle+\langle
L_{01},\hat{v}\rangle+\langle L_{11}u,\hat{v}\rangle+\frac{1}{2}\langle
L_{20}u,u\rangle+\frac{1}{2}\langle L_{02}\hat{v},\hat{v}\rangle$

where

$L_{0}=F_{000}(\tilde{x})-F_{000}(x)\ \ \ L_{1}=F_{100}(\tilde{x})-F_{100}(x)$
(4.7)
$\begin{cases}L_{10}={\underline{A_{t}}}^{T}F_{010}(\tilde{x})-F_{010}(x)-B_{t}F_{001}(x)\\\
L_{01}=C_{t}F_{010}(\tilde{x})+F_{001}(\tilde{x})-\underline{A_{t}}F_{001}(x)\end{cases}$
(4.8)
$\begin{cases}L_{11}=F_{011}(\tilde{x})\underline{A_{t}}-\underline{A_{t}}F_{011}(x)+C_{t}F_{020}(\tilde{x})\underline{A_{t}}-\underline{A_{t}}F_{002}(x)B_{t}\\\
L_{11}^{T}={\underline{A_{t}}}^{T}F_{011}^{T}(\tilde{x})-F_{011}^{T}(x){\underline{A_{t}}}^{T}+{\underline{A_{t}}}^{T}F_{020}(\tilde{x})C_{t}-B_{t}F_{002}(x){\underline{A_{t}}}^{T}\\\
L_{20}={\underline{A_{t}}}^{T}F_{020}(\tilde{x})\underline{A_{t}}-F_{020}(x)-B_{t}F_{002}(x)B_{t}-B_{t}F_{011}(x)-F_{011}^{T}(x)B_{t}\\\
L_{02}=C_{t}F_{020}(\tilde{x})C_{t}+F_{020}(\tilde{x})-\underline{A_{t}}F_{002}(x){\underline{A_{t}}}^{T}+F_{011}(\tilde{x})C_{t}+C_{t}F_{011}^{T}(\tilde{x})\end{cases}$
(4.9)

Then we solve the following equations

$\begin{cases}L_{0}=tR_{000}(x)-[tR_{000}],L_{1}=tR_{100}(x)-[tR_{100}]\\\
L_{10}=tR_{010}(x),L_{01}=tR_{001}(x)\\\
L_{11}=tR_{011}(x)-t\hat{A},L_{11}^{T}=tR_{011}^{T}(x)-t{\hat{A}}^{T},\\\
L_{20}=tR_{020}(x)-t\hat{B},L_{02}=tR_{002}(x)-t\hat{C}\end{cases}$ (4.10)

where $\hat{A},\hat{B}$ and $\hat{C}$ will be explained later, $[\cdot]$
represents the mean value. In order to avoid irreversible of the coefficient
matrix of the system, we need to satisfy the small divisor conditions listed
as:

$|\langle k,t\omega_{v}\rangle-2\pi
l|\geq\frac{t\gamma_{v}}{({1+|k|)}^{\tau}}$ (4.11) $|\langle
k,t\omega_{v}\rangle-\theta^{t,v}_{j}-2\pi
l|\geq\frac{t\gamma_{v}}{({1+|k|)}^{\tau}}$ (4.12) $|\langle
k,t\omega_{v}\rangle-\theta^{t,v}_{i}-\theta^{t,v}_{j}-2\pi
l|\geq\frac{t\gamma_{v}}{({1+|k|)}^{\tau}}$ (4.13) $|\langle
k,t\omega_{v}\rangle+\theta^{t,v}_{i}-\theta^{t,v}_{j}-2\pi
l|\geq\frac{t\gamma_{v}}{({1+|k|)}^{\tau}}$ (4.14)

for all $\ k\in\mathbb{Z}^{n}\setminus\\{0\\}$, $v=0,1,2,3,\dots$ and $i\neq
j$, where $\tau$ is a fixed integer. As for $k=0$, the corresponding small
divisor conditions should also be satisfied, as detailed in Remark 5.3 of the
measure estimation section. Then we can solve for the coefficients of $F$, and
we get $F$.

######  Remark 4.2.

Similar to [5], as $\underline{A_{t}}$, $B_{t}$, $C_{t}$ are all diagonal, the
operator $L:(F_{010},F_{001})\to(L_{10},L_{01})$ decomposes into $m$
independent operators. Now let $(M_{k})_{2\times 2}$ represent the coefficient
matrix of (4.8), similar to [5] we have
$|\det(M_{k})|\geq\frac{ct^{2}\gamma^{2}}{(|k|+1)^{2\tau}}$. By the result:
$|\det A|\leq c(n)l^{n}$ , where $A_{n\times n}=(a_{ij})$ is a matrix ,
$l=\max\limits_{1\leq i,j\leq n}|a_{ij}|$. Then, $\sum\limits_{1\leq i,j\leq
n}|(M_{k})_{ij}|\geq\max\limits_{1\leq i,j\leq
n}|(M_{k})_{ij}|\geq\frac{ct\gamma}{(|k|+1)^{\tau}}$. So similar to [5],
$\lVert X\rVert_{s-\rho}^{*}\leq\frac{c\lVert
Y\rVert_{s}^{*}}{\gamma^{\bar{n}+1}\rho^{\tau(\bar{n}+1)+\bar{n}+n}}$ with
$X=(F_{010},F_{001})^{T}$ and $Y=(R_{010},R_{001})^{T}$. The coefficient
matrix of (4.9) is similar.

Let

$\displaystyle\hat{\omega}$
$\displaystyle=diag(\hat{\omega}_{1},\dots,\hat{\omega}_{m})$
$\displaystyle\hat{A}$ $\displaystyle=diag(\hat{A}_{1},\dots,\hat{A}_{m})$
$\displaystyle\hat{B}$ $\displaystyle=diag(\hat{B}_{1},\dots,\hat{B}_{m})$
$\displaystyle\hat{C}$ $\displaystyle=diag(\hat{C}_{1},\dots,\hat{C}_{m})$

with

$\hat{\omega}_{j}=[R_{100}^{jj}]\ \ \hat{A}_{j}=[R_{011}^{jj}]\ \
\hat{B}_{j}=[R_{020}^{jj}]\ \ \hat{C}_{j}=[R_{002}^{jj}]$

Let

$\hat{N}(p,\hat{q})=\langle t\hat{\omega},\hat{y}\rangle+\langle
t\hat{A}u,\hat{v}\rangle+\frac{1}{2}\langle
t\hat{B}u,u\rangle+\frac{1}{2}\langle t\hat{C}\hat{v},\hat{v}\rangle$

Then equation (4.10) is $LF=tR-\hat{N}$, similar to [5] we have

$|\lVert
X_{F}\rVert|_{r;D(s-\rho,r)}^{*}\leq\frac{c\epsilon}{\gamma^{\bar{\nu}}\rho^{\nu}}$

where $\bar{\nu}=\bar{n}+1$ , $\nu=\tau(\bar{n}+1)+\bar{n}+n$. And $\psi$ can
be expressed as

$\psi(p,q)=(p_{+},q_{+})+X_{F}(p,q_{+})\ with\ X_{F}=(-F_{q_{+}},F_{p})$

By the same calculation method in [5] , we have $\psi:D(s-2\rho,r/2)\to
D(s-\rho,r)$, moreover

$|\lVert\psi-
id\rVert|_{r;D(s-3\rho,r/4)}^{*}\leq\frac{c\epsilon}{\gamma^{\bar{\nu}}\rho^{\nu}}\
|\lVert D\psi-
id\rVert|_{r,r;D(s-3\rho,r/4)}^{*}\leq\frac{c\epsilon}{\gamma^{\bar{\nu}}\rho^{\nu+1}}$

where $|\lVert\cdot\rVert|_{r,r}$ represent the operator norm under
$|\lVert\cdot\rVert|_{r}$.

Similar to [5] , now let $\bar{F}={\psi}^{-1}\circ F\circ\psi$, then it can be
generated by $t\bar{H}=t\bar{N}+t\bar{P}$ with $\bar{P}=\widetilde{P}+Q$, and

$t\bar{N}=tN+t\hat{N}=\langle
t\omega_{+},\hat{y}\rangle+\langle\bar{A_{t}}u,\hat{v}\rangle+\frac{1}{2}\langle\bar{B_{t}}u,u\rangle+\frac{1}{2}\langle\bar{C_{t}}\hat{v},\hat{v}\rangle$

where $t\omega_{+}=t\omega+t\hat{\omega}$ , $\bar{A_{t}}=A_{t}+t\hat{A}$,
$\bar{B_{t}}=B_{t}+t\hat{B}$, $\bar{C_{t}}=C_{t}+t\hat{C}$ with
$\lVert\hat{\omega}\rVert,\lVert\hat{A}\rVert,\lVert\hat{B}\rVert,\lVert\hat{C}\rVert\leq\epsilon$.

Note that $\underline{\bar{A_{t}}}^{2}-\bar{B_{t}}^{2}=I$ and
$\bar{B_{t}}=\bar{C_{t}}$ may no longer hold with
$\underline{\bar{A_{t}}}=I+\bar{A_{t}}$, so we need normalization. There
exists $\bar{\psi}$ symplectic, and it can transform $t\bar{N}$ into a normal
form

$tN_{+}={\bar{\psi}}^{-1}\circ\bar{F}\circ\bar{\psi}=\langle
t\omega_{+},\hat{y}\rangle+\langle
A_{t}^{+}u,\hat{v}\rangle+\frac{1}{2}\langle
B_{t}^{+}u,u\rangle+\frac{1}{2}\langle C_{t}^{+}\hat{v},\hat{v}\rangle$

where $I+A_{t}^{+}=\underline{A_{t}^{+}}=\sec\theta^{t}_{+}$,
$B_{t}^{+}=C_{t}^{+}=\tan\theta^{t}_{+}$, and
$\lVert\theta^{t}_{+}-\theta^{t}\rVert^{*}\leq ct\epsilon$. Let
$\Psi=\psi\circ\bar{\psi}$, similar to [5], $\Psi$ is generated by

$\displaystyle\langle x,y_{+}\rangle+\langle\lambda
u,v_{+}\rangle+F_{000}(x)+\langle F_{100}(x),y_{+}\rangle+\langle
F_{010}(x),u\rangle+\langle\lambda F_{001}(x),v_{+}\rangle$
$\displaystyle+\langle\lambda F_{011}(x)u,v_{+}\rangle+\frac{1}{2}\langle
F_{020}(x)u,u\rangle+\frac{1}{2}\langle\lambda\beta
v_{+},v_{+}\rangle+\frac{1}{2}\langle{\lambda}^{2}F_{002}(x)v_{+},v_{+}\rangle$

Where $\lambda=\lambda(t)$ and $\beta=\beta(t)$, please refer to the
normalization process in [5] (Lemma 3.5 in [5]), and $\lambda-I=O(\epsilon)$ ,
$\beta=O(\epsilon)$. We have $tH_{+}=tH\circ\Psi=tN_{+}+tP_{+}$, and

$|\lVert\Psi-
id\rVert|_{r;D(s-3\rho,r/4)}^{*}\leq\frac{c\epsilon}{\gamma^{\bar{\nu}}\rho^{\nu}}\
|\lVert D\Psi-
id\rVert|_{r,r;D(s-3\rho,r/4)}^{*}\leq\frac{c\epsilon}{\gamma^{\bar{\nu}}\rho^{\nu+1}}$

As for $tP_{+}$, similar to [5], let us focus on the truncated remainder
$t\widetilde{P}$ and the higher order small quantity $Q$. Then we have
$|\lVert X_{t\widetilde{P}}\rVert|_{\eta r;D(s-5\rho,\eta r)}^{*}\leq\eta
t\epsilon$, and $|\lVert
X_{Q}\rVert|_{r;D(s-5\rho,r/16)}^{*}\leq\frac{ct\epsilon^{2}}{\gamma^{2\bar{\nu}}\rho^{2\nu}}$.
As $t\bar{P}=t\widetilde{P}+Q$, we have $|\lVert X_{t\bar{P}}\rVert|_{\eta
r;D(s-5\rho,\eta r)}^{*}\leq\eta
t\epsilon+\frac{c\eta^{-2}t\epsilon^{2}}{\gamma^{2\bar{\nu}}\rho^{2\nu}}$. Let
$\eta^{3}=\frac{\epsilon}{\gamma^{\bar{\nu}}\rho^{\nu}}$, $r_{+}=\eta r$,
$s_{+}=s-5\rho$ and $\epsilon_{+}=c\eta\epsilon$, then

$|\lVert X_{tP_{+}}\rVert|_{r_{+};D(s_{+},r_{+})}^{*}\leq t\epsilon_{+}$

Consider KAM iteration, similar to [5], let
$\Psi_{0}=\Psi,\omega_{t0}=\omega_{t},\theta^{t,0}=\theta^{t}$, and
$s_{0}=s,\rho_{0}=s/20,r_{0}=r,\epsilon_{0}=\epsilon,\gamma_{0}=\gamma$,
define

$\rho_{v}=\frac{\rho_{0}}{2^{v}},\gamma_{v}=\frac{\gamma^{(\bar{n}+1)L}}{2^{(\bar{n}+1)Lv}},\eta_{v}=\frac{\epsilon_{v}}{\gamma_{v}^{2{\bar{\nu}}}\rho_{v}^{2\nu}},r_{v+1}=\eta_{v}r_{v},s_{v+1}=s_{v}-5\rho_{v}$

Where $L$ is a fixed integer, see latter Lemma 5.3, then

$\lVert t\omega_{v+1}-t\omega_{v}\rVert^{*}\leq
ct\epsilon_{v},\lVert\theta^{t,v+1}-\theta^{t,v}\rVert^{*}\leq
ct\epsilon_{v},|\lVert X_{tP_{v}}\rVert|_{r_{v};D(s_{v},r_{v})}^{*}\leq
t\epsilon_{v}$

for $\forall\ j\geq 1$. Let $F_{v+1}=({\Psi}^{v})^{-1}\circ
F_{v}\circ{\Psi}^{v}$ with
${\Psi}^{v}={\Psi}_{0}\circ{\Psi}_{1}\circ\dots\circ{\Psi}_{v}$, we have

$\displaystyle|\lVert\Psi_{v}-id\rVert|_{r_{v};D(s_{v}-3\rho_{v},r_{v})}^{*}\leq\frac{c\epsilon_{v}}{\gamma_{v}^{\bar{\nu}}{\rho_{v}}^{\nu}}$
$\displaystyle|\lVert
D\Psi_{v}-id\rVert|_{r_{v},r_{v};D(s_{v}-3\rho_{v},r_{v})}^{*}\leq\frac{c\epsilon_{v}}{\gamma_{v}^{\bar{\nu}}{\rho_{v}}^{\nu+1}}$

Set $\Psi_{\infty}=\lim\limits_{v\to\infty}{\Psi}^{v}$, let
$tH_{v}=tN_{v}+tP_{v}\to tH_{*}=tN_{*}+tP_{*}$ with

$tN_{*}=\langle t\omega_{\infty}(\xi),\hat{y}\rangle+\langle
A_{t}^{\infty}(\xi)u,\hat{v}\rangle+\frac{1}{2}\langle
B_{t}^{\infty}(\xi)u,u\rangle+\frac{1}{2}\langle
C_{t}^{\infty}(\xi)\hat{v},\hat{v}\rangle$

where $I+A_{t}^{\infty}=\underline{A_{t}^{\infty}}=\sec{\theta^{t,\infty}}$ ,
$B_{t}^{\infty}=C_{t}^{\infty}=\tan{\theta^{t,\infty}}$ . And

$\lVert t\omega_{\infty}-t\omega\rVert^{*}\leq
ct\epsilon,\lVert\theta^{t,\infty}-\theta^{t}\rVert^{*}\leq ct\epsilon$

Thus, we get a family of invariant torus for $\xi\in V_{\gamma}$. As for
measure estimation in Theorem 3.1, next section gives it.

######  Remark 4.3.

By Lemma 3.5 in [5], we have

$\underline{A_{t}^{+}}=\frac{\lambda\underline{A_{t}}}{\lambda+\beta
B_{t}},B_{t}^{+}=C_{t}^{+}=\frac{B_{t}}{\lambda(\lambda+\beta B_{t})}$

By $\lambda-I=O(\epsilon)$ , $\beta=O(\epsilon)$, for $\epsilon$ small enough
we have $\frac{|B_{t}^{+}|}{|B_{t}|}\leq\frac{1}{1-c\epsilon}$, where $c$ is a
positive constant. And by $\epsilon_{+}\sim\epsilon^{\frac{4}{3}}$, so for
$\epsilon$ small enough, we have $c\epsilon_{v}\leq\frac{1}{2^{v+2}-1}$. Then
$\frac{|B_{t}^{v+1}|}{|B_{t}^{v}|}\leq\frac{2^{v+2}-1}{2^{v+2}-2}=\frac{\sum\limits_{i=0}^{v+1}\frac{1}{2^{i}}}{\sum\limits_{i=0}^{v}\frac{1}{2^{i}}}$.
By induction, if $|B_{t}^{v}|\leq\sum\limits_{i=0}^{v}\frac{1}{2^{i}}$, then
$|B_{t}^{v+1}|\leq\sum\limits_{i=0}^{v+1}\frac{1}{2^{i}}$. So we have
$\frac{|B_{t}^{v}|}{|B_{t}^{0}|}\leq\sum\limits_{i=0}^{v}\frac{1}{2^{i}}\leq
2$, for $B_{t}^{v}=B^{v}t$, we have $|B^{v}t|\leq 2|B^{0}t|$ for
$v=0,1,2,3\dots$. Similarly, we have
$\frac{|B_{t}^{+}|}{|B_{t}|}\geq\frac{1}{1+c\epsilon}$, and we can get
$c\epsilon_{v}\leq 1$, then we have
$\frac{|B_{t}^{v+1}|}{|B_{t}^{v}|}\geq\frac{1}{2}=\frac{2-\sum\limits_{i=0}^{v+1}\frac{1}{2^{i}}}{2-\sum\limits_{i=0}^{v}\frac{1}{2^{i}}}$.
So by induction, we have $|B_{t}^{v}|\geq
2-\sum\limits_{i=0}^{v}\frac{1}{2^{i}}$, so
$\frac{|B_{t}^{v}|}{|B_{t}^{0}|}\geq
2-\sum\limits_{i=0}^{v}\frac{1}{2^{i}}\geq\frac{1}{2}$,
$|B^{v}t|\geq\frac{1}{2}|B^{0}t|$ for $v=1,2,3\dots$. This remark is in
preparation for the later measure estimation.

##  5 Measure estimation

Firstly, for all $\ k\in\mathbb{Z}^{n}\setminus\\{0\\}$ and $i\neq j$, let us
focus on the following four conditions.

$|\langle k,t\omega_{v}(\xi)\rangle-2\pi l|\geq t\gamma_{v}|k|^{-\tau}$ (5.1)
$|\langle k,t\omega_{v}(\xi)\rangle-\theta^{t,v}_{j}-2\pi l|\geq
t\gamma_{v}|k|^{-\tau}$ (5.2) $|\langle
k,t\omega_{v}(\xi)\rangle-\theta^{t,v}_{i}-\theta^{t,v}_{j}-2\pi l|\geq
t\gamma_{v}|k|^{-\tau}$ (5.3) $|\langle
k,t\omega_{v}(\xi)\rangle+\theta^{t,v}_{i}-\theta^{t,v}_{j}-2\pi l|\geq
t\gamma_{v}|k|^{-\tau}$ (5.4)

We want $t\omega_{v}$ satisfy them with corresponding $\gamma_{v}$ for
$v=0,1,2,3\dots$

Firstly, in order for $t\omega_{v},v=0,1,2,3\dots$ to satisfy (5.1), we focus
on

$\big{\\{}\xi\in V:\lvert\langle k,t\omega_{v}(\xi)\rangle-2\pi
l\rvert\geq\frac{t\gamma_{v}}{{\lvert k\rvert}^{\tau}},\forall\
k\in\mathbb{Z}^{n}\setminus\\{0\\},\forall\ l\in\mathbb{Z}\big{\\}}$

Then, we know that, if $\xi$ is a member of the set, then (5.1) holds for such
$\xi$. So we estimate its measure, and we give the following theorem.

######  Theorem 5.1.

Let $V\subseteq\mathbb{R}^{n}$ be compact , $d$ is the diameter of
$V\times(0,1)$, define
$\mathcal{B}:={V\times(0,1)+r}\subseteq\mathbb{R}^{n+1}$ for some $r>0$, and
$\omega\in C^{\bar{n}+1}(\mathcal{B},\mathbb{R})$ satisfy the Rüssmann’s non-
degenerate condition with the index $\bar{n}$ and amount $\beta$. Then for
aforementioned $\omega_{v},v=0,1,2,3\dots$ , let

$\bar{V}_{v}=\big{\\{}\xi\in V:|\langle k,t\omega_{v}\rangle-2\pi
l|\geq\frac{t\gamma_{v}}{|k|^{\tau}},\forall\
k\in\mathbb{Z}^{n}\setminus\\{0\\},\forall\ l\in\mathbb{Z}\big{\\}}$ (5.5)
$\bar{V}=\bigcap\limits_{v=0}^{\infty}\bar{V}_{v}$ (5.6)

then we have the estimate

$\big{|}V\setminus\bar{V}\big{|}\leq c_{3}\gamma
d^{n}((n+1)^{-\frac{1}{2}}+2d+\theta^{-1}d)$

where $c_{3}=12\bar{c}(2\pi
e)^{\frac{n+1}{2}}(\bar{n}+2)^{\bar{n}+3}[(\bar{n}+2)!]^{-1}\big{(}\lvert\omega\rvert_{V}+c\epsilon+1\big{)}\beta^{-\frac{\bar{n}+2}{\bar{n}+1}}\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}$,
$\bar{c}$ depends only on $n$, $\bar{n}$ and $\tau$. And for $\gamma$ small
enough , $\bar{V}$ can be a set of positive measures.

Before proving the theorem, we first refer to the following lemma in [4] ).

######  Lemma 5.1.

(Lemma 4.9 in [4]) Let $K\subseteq\mathbb{R}^{n}$ be compact with positive
diameter $d:=\sup_{x,y\in K}|x-y|_{2}>0$, define
$\mathcal{B}:={K+\theta}\subseteq\mathbb{R}^{n}$ for some $\theta>0$, and
$g\in C^{u_{0}+1}(\mathcal{B},\mathbb{R})$ be a function with

$\min_{y\in K}\max_{0\leq v\leq{u_{0}}}|D^{v}g(y)|\geq\beta$ (5.7)

for some ${u_{0}}\in\mathbb{N}$ and $\beta>0$. Then for any $\tilde{g}\in
C^{u_{0}}(\mathcal{B},\mathbb{R})$ satisfying
$|\tilde{g}-g|^{u_{0}}_{\mathcal{B}}\leq\frac{1}{2}\beta$ we have the estimate

$\big{|}\big{\\{}y\in K:|\tilde{g}(y)|\leq\epsilon\big{\\}}\big{|}\leq
Bd^{n-1}(n^{-\frac{1}{2}}+2d+\theta^{-1}d)(\frac{\epsilon}{\beta})^{\frac{1}{u_{0}}}\frac{1}{\beta}\max_{0<v\leq{u_{0}}}|D^{v}g|_{\mathcal{B}}$
(5.8)

whenever $0<\epsilon<\frac{\beta}{2u_{0}+2}$. Here, $B:=3(2\pi
e)^{\frac{n}{2}}(u_{0}+1)^{u_{0}+2}[(u_{0}+1)!]^{-1}$.

######  Remark 5.1.

This Lemma is Lemma 4.9 in [4], which comes from Theorem 17.1 in [3], and the
proof is similar to the proof of Theorem 17.1 in [3].

Now, let us focus on the proof of Theorem 5.1.

#### Proof of Theorem 5.1

Firstly, for $t\omega$ satisfies the Rüssmann’s non-degenerate condition, by
Remark 2.1, $\exists\ \bar{n}=\bar{n}(\omega,V)\in\mathbb{N}$ and
$\beta=\beta(\omega,V)>0$ such that $\min\limits_{\xi\in V}\max\limits_{0\leq
v\leq\bar{n}}\lvert D^{v}\langle k,t\omega(\xi)\rangle\rvert\geq t\beta\lvert
k\rvert$ for $\forall\ k\in\mathbb{Z}^{n}\setminus\\{0\\}$.

Let $\widetilde{V}=V\times(0,1)$, $\widetilde{\xi}=(\xi,\xi^{{}^{\prime}})$,
$\xi^{{}^{\prime}}\in(0,1)$, $t\widetilde{\omega}=(t\omega,-2\pi)$,
$\widetilde{k}=(k,l)$, then we have
$rank\big{\\{}\partial^{i}_{\xi}\widetilde{\omega}(\widetilde{\xi}):|i|\leq\bar{n}+1\big{\\}}=n+1,\
\widetilde{\xi}\in\widetilde{V}$, and we can easily get that
$\min\limits_{\widetilde{\xi}\in\widetilde{V}}\max\limits_{0\leq
v\leq\bar{n}+1}|D^{v}\langle\widetilde{k},t\widetilde{\omega}(\widetilde{\xi})\rangle|\geq
t\beta\lvert\widetilde{k}\rvert$.

Let $\mathcal{B}=(\widetilde{V}+r)\cap\mathbb{R}^{n+1}$, by
$\lVert\omega_{v}-\omega\rVert^{*}\leq c\epsilon$, then for $\epsilon$ small
enough, we have
$\lvert\langle\widetilde{k},t\widetilde{\omega}_{v}\rangle-\langle\widetilde{k},t\widetilde{\omega}\rangle\rvert^{\bar{n}+1}_{\mathcal{B}}\leq\frac{1}{2}t\beta\lvert\widetilde{k}\rvert$,
and for $\gamma_{0}$ small enough, we have $t\gamma_{v}\lvert
k\rvert^{-\tau}\leq\frac{t\beta\lvert\widetilde{k}\rvert}{2(\bar{n}+1)+2}$, so
we can use Lemma 5.1, and we have

$\displaystyle\big{\lvert}\\{\xi\in V:\lvert\langle k,t\omega_{v}\rangle-2\pi
l\rvert\leq t\gamma_{v}\lvert
k\rvert^{-\tau}\\}\big{\rvert}=\big{\lvert}\\{\widetilde{\xi}\in\widetilde{V}:\lvert\langle\widetilde{k},t\widetilde{\omega}_{v}\rangle\rvert\leq
t\gamma_{v}\lvert k\rvert^{-\tau}\\}\big{\rvert}$ $\displaystyle\ \leq\
c_{1}d^{n}((n+1)^{-\frac{1}{2}}+2d+r^{-1}d)(\frac{t\gamma_{v}}{t\beta})^{\frac{1}{\bar{n}+1}}\lvert
k\rvert^{\frac{-\tau}{\bar{n}+1}}\lvert\widetilde{k}\rvert^{\frac{-1}{\bar{n}+1}}\frac{1}{t\beta}\max_{0<v\leq\bar{n}+2}|D^{v}t\widetilde{\omega}|_{\mathcal{B}}$

for $k\in\mathbb{Z}^{n}\setminus\\{0\\},l\in\mathbb{Z}$ , where $c_{1}=3(2\pi
e)^{\frac{n+1}{2}}(\bar{n}+2)^{\bar{n}+3}[(\bar{n}+2)!]^{-1}$, $d$ is the
diameter of $V\times(0,1)$. For such $l$ right here, we have $\lvert
l\rvert\leq\lvert k\rvert\hat{K}$, where
$\hat{K}=\lvert\omega\rvert_{V}+c\epsilon+1$. Now we define

$\displaystyle R^{k,l}_{v}$ $\displaystyle=\big{\\{}\xi\in V:\lvert\langle
k,t\omega_{v}\rangle-2\pi l\rvert<t\gamma_{v}\lvert k\rvert^{-\tau}\big{\\}},\
\lvert l\rvert\leq\lvert k\rvert\hat{K},\ k\in\mathbb{Z}^{n}\setminus\\{0\\}$
$\displaystyle R^{k}_{v}$ $\displaystyle=\big{\\{}\xi\in V:\lvert\langle
k,t\omega_{v}\rangle-2\pi l\rvert<t\gamma_{v}\lvert k\rvert^{-\tau},\exists\
l\in\mathbb{Z}\big{\\}},\ k\in\mathbb{Z}^{n}\setminus\\{0\\}$ $\displaystyle
R_{v}$ $\displaystyle=\big{\\{}\xi\in V:\lvert\langle
k,t\omega_{v}\rangle-2\pi l\rvert<t\gamma_{v}\lvert k\rvert^{-\tau},\exists\
l\in\mathbb{Z},\exists\ k\in\mathbb{Z}^{n}\setminus\\{0\\}\big{\\}}$

i.e.,
$R_{v}=\bigcup\limits_{k\in\mathbb{Z}^{n}\setminus\\{0\\}}R^{k}_{v}=\bigcup\limits_{k\in\mathbb{Z}^{n}\setminus\\{0\\}}\bigcup\limits_{\lvert
l\rvert\leq\lvert k\rvert\hat{K}}R^{k,l}_{v}$. Besides, for
$k\in\mathbb{Z}^{n}\setminus\\{0\\}$ we have
$\lvert\widetilde{k}\rvert^{\frac{-1}{\bar{n}+1}}\leq 1$ , then,

$\displaystyle\lvert R^{k,l}_{v}\rvert$ $\displaystyle\leq
c_{1}d^{n}((n+1)^{-\frac{1}{2}}+2d+r^{-1}d)(\frac{t\gamma_{v}}{t\beta})^{\frac{1}{\bar{n}+1}}\frac{1}{t\beta}\max_{0<v\leq\bar{n}+2}|D^{v}t\widetilde{\omega}|_{\mathcal{B}}\lvert
k\rvert^{\frac{-\tau}{\bar{n}+1}}$ $\displaystyle\leq
c_{1}d^{n}((n+1)^{-\frac{1}{2}}+2d+r^{-1}d)(\frac{\gamma_{v}}{\beta})^{\frac{1}{\bar{n}+1}}\frac{1}{\beta}\max_{0<v\leq\bar{n}+2}|D^{v}\omega|_{\mathcal{B}}\lvert
k\rvert^{\frac{-\tau}{\bar{n}+1}}$ $\displaystyle\lvert R^{k}_{v}\rvert$
$\displaystyle=\lvert\bigcup\limits_{\lvert l\rvert\leq\lvert
k\rvert\hat{K}}R^{k,l}_{v}\rvert\leq 2\lvert k\rvert\hat{K}\lvert
R^{k,l}_{v}\rvert$ $\displaystyle\leq
2\hat{K}c_{1}d^{n}((n+1)^{-\frac{1}{2}}+2d+r^{-1}d)(\frac{\gamma_{v}}{\beta})^{\frac{1}{\bar{n}+1}}\frac{1}{\beta}\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}\lvert
k\rvert^{\frac{\bar{n}+1-\tau}{\bar{n}+1}}$ $\displaystyle\leq
c_{2}d^{n}((n+1)^{-\frac{1}{2}}+2d+r^{-1}d)(\frac{\gamma_{v}}{\beta})^{\frac{1}{\bar{n}+1}}\frac{1}{\beta}\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}\lvert
k\rvert^{\frac{\bar{n}+1-\tau}{\bar{n}+1}}$

where $c_{2}=6(2\pi
e)^{\frac{n+1}{2}}(\bar{n}+2)^{\bar{n}+3}[(\bar{n}+2)!]^{-1}\hat{K}$.

By (5.6) $\bar{V}=\bigcap\limits_{v=0}^{\infty}\bar{V}_{v}$, we have the
following inequality.

$\displaystyle\lvert V\setminus\bar{V}\rvert\leq\sum_{v=0}^{\infty}\lvert
R_{v}\rvert\
\leq\sum_{v=0}^{\infty}\sum_{k\in\mathbb{Z}^{n}\setminus\\{0\\}}\lvert
R_{v}^{k}\rvert\leq\sum_{v=0}^{\infty}\sum^{\infty}_{r=1}2n(2r+1)^{n-1}\lvert
R_{v}^{r}\rvert$
$\displaystyle\leq\sum_{v=0}^{\infty}\sum^{\infty}_{r=1}2n(2r+1)^{n-1}c_{2}d^{n}((n+1)^{-\frac{1}{2}}+2d+\theta^{-1}d)(\frac{\gamma_{v}}{\beta})^{\frac{1}{\bar{n}+1}}\frac{1}{\beta}\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}r^{\frac{\bar{n}+1-\tau}{\bar{n}+1}}$
$\displaystyle\leq\sum^{\infty}_{r=1}2n(2r+1)^{n-1}r^{\frac{\bar{n}+1-\tau}{\bar{n}+1}}c_{2}d^{n}((n+1)^{-\frac{1}{2}}+2d+\theta^{-1}d)$
$\displaystyle\
\cdot\sum_{v=0}^{\infty}\gamma_{v}^{\frac{1}{\bar{n}+1}}\beta^{-\frac{\bar{n}+2}{\bar{n}+1}}\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}$

As for $\tau\geq(n+2)(\bar{n}+1)$,
$\bar{c}:=\sum\limits_{r=1}^{\infty}2n(2r+1)^{n-1}r^{\frac{\bar{n}+1-\tau}{\bar{n}+1}}$
is convergent. Besides,
$\gamma_{v}\leq\frac{\gamma^{\bar{n}+1}}{2^{(\bar{n}+1)v}}$, we have
$\sum\limits_{v=0}^{\infty}\gamma_{v}^{\frac{1}{\bar{n}+1}}\leq\sum\limits_{v=0}^{\infty}\frac{\gamma}{2^{v}}=2\gamma$,
and $\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}<\infty$ by (2.3). Therefore,
we have

$\displaystyle\lvert V\setminus\bar{V}\rvert$ $\displaystyle\leq\bar{c}\cdot
2\gamma\cdot
c_{2}d^{n}((n+1)^{-\frac{1}{2}}+2d+\theta^{-1}d)\beta^{-\frac{\bar{n}+2}{\bar{n}+1}}\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}$
$\displaystyle\leq c_{3}\gamma d^{n}((n+1)^{-\frac{1}{2}}+2d+\theta^{-1}d)$

where $c_{3}=12\bar{c}(2\pi
e)^{\frac{n+1}{2}}(\bar{n}+2)^{\bar{n}+3}[(\bar{n}+2)!]^{-1}\big{(}\lvert\omega\rvert_{V}+c\epsilon+1\big{)}\beta^{-\frac{\bar{n}+2}{\bar{n}+1}}\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}$,
$\bar{c}$ depends only on $n$, $\bar{n}$ and $\tau$. So, for $\gamma$ small
enough , $\bar{V}$ can be a set of positive measures.

The proof of Theorem 5.1 is complete. $\square$

So we get the measure estimate that $t\omega_{v},v=0,1,2,3\dots$ satisfy
(5.1).

Now, let us focus on (5.2). By the way, (5.3) and (5.4) are similar to that.
Firstly, similar to (5.1), we focus on

$\big{\\{}\xi\in V:\lvert\langle
k,t\omega_{v}(\xi)\rangle-\theta_{j}^{t,v}-2\pi
l\rvert\geq\frac{t\gamma_{v}}{{\lvert k\rvert}^{\tau}},\forall\
k\in\mathbb{Z}^{n}\setminus\\{0\\},\forall\ l\in\mathbb{Z}\big{\\}}$

We estimate its measure, and we give the following theorem.

######  Theorem 5.2.

For the same notations in Theorem 5.1, and
$\theta_{j}^{t,v}(\xi),j=1,2,\dots,m$ are the ones above, let

$\bar{\bar{V}}_{v}=\big{\\{}\xi\in V:|\langle
k,t\omega_{v}\rangle-\theta_{j}^{t,v}-2\pi
l|\geq\frac{t\gamma_{v}}{|k|^{\tau}},\forall
k\in\mathbb{Z}^{n}\setminus\\{0\\},\forall l\in\mathbb{Z},\forall 1\leq j\leq
m\big{\\}}$ $\bar{\bar{V}}=\bigcap\limits_{v=1}^{\infty}\bar{\bar{V}}_{v}$
(5.9)

then we have the estimate

$\big{|}V\setminus\bar{\bar{V}}\big{|}\leq c_{5}\gamma d^{n}$ (5.10)

here
$c_{5}=4\bar{\bar{c}}c_{4}m(\lvert\omega\rvert_{V}+c\epsilon+\max\limits_{1\leq
j\leq m}3|B^{0}_{j}|+1)$, $\bar{\bar{c}}$ depends only on $n$, $\bar{n}$ and
$\tau$ , $c_{4}$ and $c$ are independent of $t$, $\epsilon$ and $\gamma$. And
for $\gamma$ small enough , $\bar{\bar{V}}$ can be a set of positive measures.

#### Proof of Theorem 5.2

Review the previous, $\underline{A_{t}}=I+t(A-I),B_{t}=tB$. We need adjust
them to satisfy ${\underline{A_{t}}}^{2}-{B_{t}}^{2}=I$. Here
$\underline{A_{tj}}=\sec\theta^{t}_{j}$, $B_{tj}=\tan\theta^{t}_{j}$ for
$j=1,2,\dots,m$. When $t\to 0$, we have $\theta^{t}_{j}(\xi)\to 0$, and it can
be controlled by $3|B^{0}_{j}t|$. This is due to

$\lim_{t\to 0}\frac{|\theta_{j}^{t}|}{t}=\lim_{t\to
0}\frac{\arctan{(|tB_{j}|)}}{t}=\lim_{t\to
0}\frac{|B_{j}|}{t^{2}{B_{j}}^{2}+1}=|B_{j}|$ (5.11)

So for $t$ small enough, we have $|\theta_{j}^{t}|\leq\frac{3}{2}|B_{j}t|$,
namely, $|\theta_{j}^{t,v}|\leq\frac{3}{2}|B_{j}^{v}t|$. And from the Remark
4.3 , $|B^{v}t|\leq 2|B^{0}t|$, namely, $|B^{v}_{j}t|\leq 2|B^{0}_{j}t|$ .
Thus $|\theta^{t,v}_{j}|\leq 3|B^{0}_{j}t|$, namely, $|\theta^{t,v}|\leq
3|B^{0}t|$. Then, to complete the proof, we give some lemmas.

######  Lemma 5.2.

Suppose function $g(x)$ is real analytic on the closure $\bar{I}$ of $I$,
where $I\subset\mathbb{R}$ is an internal. $\lvert g^{(m)}(x)\rvert\geq d$ on
$I$, where $d>0$ is a constant. Let $I_{h}=\\{x\in I:\lvert g(x)\rvert<h$,
then we have $meas(I_{h})\leq 2(\frac{m!h}{d})^{\frac{1}{m}}$.

#### Proof

Note that $I_{h}$ has at most one connected component, we can assume that
$\bar{I_{h}}=[x_{0}-\sigma_{0},x_{0}+\sigma_{0}]$ with $\sigma_{0}\geq 0$.
Then, if $\sigma_{0}>0$, we have $d(x_{0},\partial I_{h})>\sigma$ for
$\forall\ 0<\sigma<\sigma_{0}$, then on $I_{h}$, by Cauchy inequality, we have
$d\leq\lvert g^{m}(x)\rvert\leq\frac{m!h}{\sigma^{m}}$. By it holds for
$\forall\ 0<\sigma<\sigma_{0}$, so $\lvert I_{h}\rvert\leq 2\sigma_{0}\leq
2(\frac{m!h}{d})^{\frac{1}{m}}$. If $\sigma_{0}=0$, the result still holds.
$\square$

Next, refer to Theorem 1.2 in [6] given by Zhang D, Wang S and Xu J, we get
the following lemma.

######  Lemma 5.3.

Suppose $\omega(\xi)$ is real analytic on $\Sigma$ and satisfy the Rüssmann’s
non-degenerate condition: $\langle\omega,y\rangle\not\equiv 0$ for all
$y\in\mathbb{R}^{n}\setminus\\{0\\}$ . Let $P(\xi)=(P_{ij}(\xi))_{N\times N}$
be an $N\times N$ matrix, and let $M$ be a matrix with

$M(\xi)=\ \langle k,t\omega(\xi)\rangle
I_{N}+diag(\theta^{t}_{1}(\xi),\theta^{t}_{2}(\xi),\dots,\theta^{t}_{N}(\xi))+P_{t}(\xi)$
(5.12)

where $I_{N}$ denotes unit matrix, $0<t<1$. Let

$R_{k}^{t}(\alpha)=\big{\\{}\xi\in\Sigma:\lVert
M(\xi)^{-1}\rVert>\frac{|k|^{\tau}}{t\alpha}\big{\\}}$ (5.13)

where $\lVert\cdot\rVert$ denotes the sup-norm of matrix. Then, there exist
sufficiently large integers $L>0$ and $K_{0}>0$ such that, if
$\theta^{t}_{j}(\xi)$ is $L$-th differentiable with
$\lVert\theta^{t}_{j}(\xi)\rVert^{L}\leq\sigma(j=1,2,\dots,N)$ and

${\lVert P_{t}(\xi)\rVert}^{L}_{\Sigma}=\sup_{\xi\in\Sigma}\max_{1\leq i,j\leq
N}{\lVert P^{t}_{i,j}(\xi)\rVert}^{L}\leq\epsilon$ (5.14)

where $\sigma$ is a positive constant, then, for sufficiently small
$\epsilon>0$ and $\alpha>0$, we have

$meas(R_{k}^{t}(\alpha))\leq
ct^{-\frac{N-1}{L}}[diam(\Sigma)]^{n-1}(\frac{\alpha}{|k|^{\tau-N}})^{\frac{1}{L}},\forall\
|k|>K_{0}$ (5.15)

where $\tau>nL+N$ is a fixed constant and $c$ is independent of $t$,
$\epsilon$ and $\alpha$.

######  Remark 5.2.

The proof of this lemma is exactly like the proof of Theorem 1.2 in [6],
except that Lemma 5.2 in this paper is used instead of Lemma 2.2 in [6].
Besides, as $c$ is independent of $\epsilon$, we can replace condition
${\lVert P_{t}(\xi)\rVert}^{L}_{\Sigma}\leq\epsilon$ with ${\lVert
P_{t}(\xi)\rVert}^{L}_{\Sigma}\leq\lvert k\rvert\epsilon$, the conclusion
still holds for fixed $k$. See [6] for specific proof.

Extend the notation in the proof of Theorem 5.1, e.g., $d$, $\bar{n}$,
$\beta$, $\widetilde{V}=V\times(0,1)$,
$\widetilde{\xi}=(\xi,\xi^{{}^{\prime}})$,
$t\widetilde{\omega}=(t\omega,-2\pi)$, $\widetilde{k}=(k,l)$. Now we let
$P_{v,l}^{t}(\widetilde{\xi})=\langle\widetilde{k},t\widetilde{\omega}_{v}(\widetilde{\xi})-t\widetilde{\omega}(\widetilde{\xi})\rangle$be
an $1\times 1$ matrix. For $\forall\ j\in\\{1,2,\dots,m\\}$, let

$M_{v,l}^{t}(\widetilde{\xi})=\langle\widetilde{k},t\widetilde{\omega}(\widetilde{\xi})\rangle-\theta_{j}^{t,v}(\widetilde{\xi})+P_{v,l}^{t}(\widetilde{\xi})$

Then, by Lemma 5.3 and Remark 5.2, we know that there are such $L>0$ and
$K_{0}>0$. From the previous analysis, $|\theta^{t,v}_{j}|\leq 3|B^{0}_{j}|t$,
it is easy to get $\lVert\theta_{j}^{t,v}\rVert^{L}\leq\sigma(j=1,2,\dots,m)$
with $\sigma$ is a positive constant. And by $\lVert
t\omega_{v}-t\omega\rVert^{*}\leq ct\epsilon$, we have ${\lVert
P_{v,l}^{t}(\widetilde{\xi})\rVert}^{L}_{\widetilde{V}}\leq
c^{{}^{\prime}}t|\widetilde{k}|\epsilon$ with $\epsilon$ small enough. Then
for $\gamma_{v}$ small enough, we have

$\lvert\big{\\{}\widetilde{\xi}\in\widetilde{V}:\lVert
M_{v}^{t}(\widetilde{\xi})^{-1}\rVert>\frac{|\widetilde{k}|^{\tau}}{t\gamma_{v}}\big{\\}}\rvert\leq
c_{4}d^{n}(\frac{\gamma_{v}}{|\widetilde{k}|^{\tau-1}})^{\frac{1}{L}},\forall\
|\widetilde{k}|=\lvert k\rvert+\lvert l\rvert>K_{0}$ (5.16)

where $\tau>(n+1)L+1$, $d$ is the diameter of $V\times(0,1)$. Similar to the
proof of Theorem 5.1, we have $\lvert l\rvert\leq\lvert
k\rvert\hat{K}^{{}^{\prime}}$ with
$\hat{K}^{{}^{\prime}}=\lvert\omega\rvert_{V}+c\epsilon+\max\limits_{1\leq
j\leq m}3|B^{0}_{j}|+1$.

Define

$\displaystyle R^{{}^{\prime}}_{k,v,l}$ $\displaystyle=\big{\\{}\xi\in
V:\lvert\langle k,t\omega_{v}\rangle-\theta_{j}^{t,v}-2\pi
l\rvert<\frac{t\gamma_{v}}{\lvert k\rvert^{\tau}}\big{\\}},\ \lvert
l\rvert\leq\lvert k\rvert\hat{K}^{{}^{\prime}},\ \lvert k\rvert>K_{0}$
$\displaystyle R^{{}^{\prime}}_{k,v}$ $\displaystyle=\big{\\{}\xi\in
V:\lvert\langle k,t\omega_{v}\rangle-\theta_{j}^{t,v}-2\pi
l\rvert<\frac{t\gamma_{v}}{\lvert k\rvert^{\tau}},\exists\
l\in\mathbb{Z}\big{\\}},\ \lvert k\rvert>K_{0}$ $\displaystyle
R^{{}^{\prime}}_{v}$ $\displaystyle=\big{\\{}\xi\in V:\lvert\langle
k,t\omega_{v}\rangle-\theta_{j}^{t,v}-2\pi l\rvert<\frac{t\gamma_{v}}{\lvert
k\rvert^{\tau}},\exists\ l\in\mathbb{Z},\exists\ \lvert
k\rvert>K_{0}\big{\\}}$

i.e., $R^{{}^{\prime}}_{v}=\bigcup\limits_{\lvert
k\rvert>K_{0}}R^{{}^{\prime}}_{k,v}=\bigcup\limits_{\lvert
k\rvert>K_{0}}\bigcup\limits_{\lvert l\rvert\leq\lvert
k\rvert\hat{K}^{{}^{\prime}}}R^{{}^{\prime}}_{k,v,l}$.

Then we have

$\lvert R^{{}^{\prime}}_{k,v,l}\rvert\leq
c_{4}d^{n}(\frac{\gamma_{v}}{|\widetilde{k}|^{\tau-1}})^{\frac{1}{L}}\leq
c_{4}d^{n}\gamma_{v}^{\frac{1}{L}}\lvert k\rvert^{-\frac{\tau-1}{L}}$

By $R^{{}^{\prime}}_{k,v}=\bigcup\limits_{\lvert l\rvert\leq\lvert
k\rvert\hat{K}}R^{{}^{\prime}}_{k,v,l}$, we have

$\lvert R^{{}^{\prime}}_{k,v}\rvert\leq 2\lvert
k\rvert\hat{K}^{{}^{\prime}}\lvert R^{{}^{\prime}}_{k,v,l}\rvert\leq
2c_{4}\hat{K}^{{}^{\prime}}d^{n}\gamma_{v}^{\frac{1}{L}}\lvert
k\rvert^{-\frac{\tau-L-1}{L}}$

Let

$\displaystyle\bar{\bar{V}}_{j,v}^{{}^{\prime}}$
$\displaystyle=\big{\\{}\xi\in V:|\langle
k,t\omega_{v}\rangle-\theta_{j}^{t,v}-2\pi
l|\geq\frac{t\gamma_{v}}{|k|^{\tau}},\forall\ |k|>K_{0},\forall\
l\in\mathbb{Z}\big{\\}}$ $\displaystyle\bar{\bar{V}}^{{}^{\prime}}_{j}$
$\displaystyle=\bigcap\limits_{v=0}^{\infty}\bar{\bar{V}}_{j,v}^{{}^{\prime}},\
\bar{\bar{V}}^{{}^{\prime}}=\bigcap\limits_{j=1}^{m}\bar{\bar{V}}^{{}^{\prime}}_{j}$
$\displaystyle\bar{\bar{V}}^{{}^{\prime\prime}}_{k,v,l,j}$
$\displaystyle=\big{\\{}\xi\in V:|\langle
k,t\omega_{v}\rangle-\theta_{j}^{t,v}-2\pi
l|\geq\frac{t\gamma_{v}}{|k|^{\tau}}\big{\\}},\ 0<|k|\leq K_{0},\
l\in\mathbb{Z}$ $\displaystyle\bar{\bar{V}}^{{}^{\prime\prime}}$
$\displaystyle=\bigcap\limits_{k\in\mathbb{Z}^{n}\setminus\\{0\\}}\bigcap\limits_{l\in\mathbb{Z}}\bigcap\limits_{j=1}^{m}\bigcap\limits_{v=1}^{\infty}\bar{\bar{V}}^{{}^{\prime\prime}}_{k,v,l,j}$

then we have the following inequality.

$\displaystyle\lvert V\setminus\bar{\bar{V}}^{{}^{\prime}}_{j}\rvert$
$\displaystyle\leq\sum_{v=0}^{\infty}\lvert R_{v}^{{}^{\prime}}\rvert\
\leq\sum_{v=0}^{\infty}\sum_{\lvert k\rvert>K_{0}}\lvert
R_{k,v}^{{}^{\prime}}\rvert$
$\displaystyle\leq\sum_{v=0}^{\infty}\sum^{\infty}_{r=K_{0}+1}2n(2r+1)^{n-1}\lvert
R_{r,v}^{{}^{\prime}}\rvert$
$\displaystyle\leq\sum_{v=0}^{\infty}\sum^{\infty}_{r=K_{0}+1}2n(2r+1)^{n-1}\cdot
2c_{4}\hat{K}^{{}^{\prime}}d^{n}\gamma_{v}^{\frac{1}{L}}r^{-\frac{\tau-L-1}{L}}$
$\displaystyle\leq\sum^{\infty}_{r=1}2n(2r+1)^{n-1}r^{-\frac{\tau-L-1}{L}}2c_{4}\hat{K}^{{}^{\prime}}d^{n}\sum_{v=0}^{\infty}\gamma_{v}^{\frac{1}{L}}$

For $\tau\geq(n+2)L+1$,
$\bar{\bar{c}}:=\sum^{\infty}_{r=1}2n(2r+1)^{n-1}r^{-\frac{\tau-L-1}{L}}$ is
convergent. And by $\gamma_{v}\leq\frac{\gamma^{L}}{2^{Lv}}$, we have
$\sum\limits_{v=0}^{\infty}\gamma_{v}^{\frac{1}{L}}\leq\sum\limits_{v=0}^{\infty}\frac{\gamma}{2^{v}}=2\gamma$.
Therefore, we have

$\lvert V\setminus\bar{\bar{V}}^{{}^{\prime}}_{j}\rvert\leq\bar{\bar{c}}\cdot
2\gamma\cdot 2c_{4}\hat{K}^{{}^{\prime}}d^{n}$

By $j$ is arbitrary and
$\bar{\bar{V}}^{{}^{\prime}}=\bigcap\limits_{j=1}^{m}\bar{\bar{V}}^{{}^{\prime}}_{j}$,
then we have

$\lvert V\setminus\bar{\bar{V}}^{{}^{\prime}}\rvert\leq m\cdot
4\bar{\bar{c}}c_{4}\hat{K}^{{}^{\prime}}\gamma d^{n}\leq c_{5}\gamma d^{n}$

where
$c_{5}=4\bar{\bar{c}}c_{4}m(\lvert\omega\rvert_{V}+c\epsilon+\max\limits_{1\leq
j\leq m}3|B^{0}_{j}|+1)$, $c_{4}$ comes from Lemma 5.3 and it is independent
of $t$, $\epsilon$ and $\gamma$. So, for $\gamma$ small enough ,
$\bar{V}^{{}^{\prime}}$ can be a set of positive measures.

So we get the measure estimate that $\omega_{v}$ satisfies the condition (5.2)
for $|k|>K_{0}$.

Next, we consider $0<|k|\leq K_{0}$. It is mainly finite thought.

Firstly, for $0<|k|\leq K_{0}$, there are only a finite number of $k$ ’s like
this. If $\lvert\langle k,t\omega_{v}(\xi)\rangle-\theta_{j}^{t,v}(\xi)-2\pi
l\rvert\geq\alpha>0$ on $\bar{\bar{V}}_{*}$ for $\forall\ 0<|k|\leq K_{0}$,
$\forall\ \lvert l\rvert\leq K_{0}\hat{K}^{{}^{\prime}}$ , $\forall\ 1\leq
j\leq m$ and $\forall\ v\in\mathbb{N}$, where
$\hat{K}^{{}^{\prime}}=\lvert\omega\rvert_{V}+c\epsilon+\max\limits_{1\leq
j\leq m}3|B^{0}_{j}|+1$, then by finiteness, we can choose the smallest
$\alpha$ like this. As for $\gamma_{v}$ small enough, we can get
$\alpha>t\gamma_{v}>\frac{t\gamma_{v}}{|k|^{\tau}}$. Then $\lvert\langle
k,t\omega_{v}(\xi)\rangle-\theta_{j}^{t,v}(\xi)-2\pi
l\rvert\geq\frac{t\gamma_{v}}{|k|^{\tau}}$ is satisfied on
$\bar{\bar{V}}^{{}^{\prime\prime}}$. So under the circumstance, we can get
that $\bar{\bar{V}}_{*}\subseteq\bar{\bar{V}}^{{}^{\prime\prime}}$.

Secondly, let us deal with the case where this condition is not met. Namely,
$\exists\ 0<|k|\leq K_{0}$, $\exists\ \lvert l\rvert\leq
K_{0}\hat{K}^{{}^{\prime}}$, $\exists\ 1\leq j\leq m$ and $\exists\
v\in\mathbb{N}$, there is no such $\alpha$ that $\lvert\langle
k,t\omega_{v}(\xi)\rangle-\theta_{j}^{t,v}(\xi)-2\pi l\rvert\geq\alpha$ on
$V\setminus\bar{\bar{V}}_{*}$. Let
$R^{{}^{\prime\prime}}_{k,v,l,j}=\big{\\{}\xi\in V:\lvert\langle
k,t\omega_{v}(\xi)\rangle-\theta_{j}^{t,v}(\xi)-2\pi l\rvert=0\big{\\}}$, by
Lemma 18.6 in [3], we have
$\big{\lvert}R^{{}^{\prime\prime}}_{k,v,l,j}\big{\rvert}=0$, so $\lvert
V\setminus\bar{\bar{V}}_{*}\rvert=0$, then we have $\lvert
V\setminus\bar{\bar{V}}^{{}^{\prime\prime}}\rvert=0$.

This is the measure estimate that $\omega_{v}$ satisfies the condition (5.2)
for $0<|k|\leq K_{0}$.

As
$\bar{\bar{V}}=\bar{\bar{V}}^{{}^{\prime}}\cap\bar{\bar{V}}^{{}^{\prime\prime}}$
, we have

$\lvert V\setminus\bar{\bar{V}}\rvert\leq\lvert
V\setminus\bar{\bar{V}}^{{}^{\prime}}\rvert+\lvert
V\setminus\bar{\bar{V}}^{{}^{\prime\prime}}\rvert\leq c_{5}\gamma d^{n}$
(5.17)

$c_{5}=4\bar{\bar{c}}c_{4}m(\lvert\omega\rvert_{V}+c\epsilon+\max\limits_{1\leq
j\leq m}3|B^{0}_{j}|+1)$, $\bar{\bar{c}}$ depends only on $n$, $\bar{n}$ and
$\tau$ , $c_{4}$ and $c$ are independent of $t$, $\epsilon$ and $\gamma$. So,
for $\gamma$ small enough , $\bar{\bar{V}}$ can be a set of positive measures.
The proof of Theorem 5.2 is complete. $\square$

So we get the measure estimate that $t\omega_{v},v=0,1,2,3\dots$ satisfy
(5.2).

As for (5.3) and (5.4), we can do a similar thing. By the previous analysis in
the proof of Theorem 5.2, we know that $|\theta^{t,v}_{j}|\leq 3|B^{0}_{j}t|$,
further to have
$\lVert\theta^{t,v}_{j}(\xi)\rVert^{L}\leq\sigma(j=1,2,\dots,m,v\in\mathbb{N})$.
So we can easily get that
$\lVert\theta^{t,v}_{i}(\xi)\pm\theta^{t,v}_{j}(\xi)\rVert^{L}\leq\sigma(j=1,2,\dots,m,v\in\mathbb{N})$,
then (5.3) and (5.4) are naturally satisfied. Then we get the following
theorem.

######  Theorem 5.3.

For the same notations in Theorem 5.1 and Theorem 5.2, assume that (5.1),
(5.2),(5.3) and (5.4) satisfied on $V_{\gamma}$ for $\forall\
k\in\mathbb{Z}^{n}\setminus\\{0\\}$, $\forall\ l\in\mathbb{Z}$ , $\forall\
1\leq j\leq m$ and $\forall\ v\in\mathbb{N}$, then we have the estimate

$\lvert V\setminus V_{\gamma}\rvert\leq c_{3}\gamma
d^{n}((n+1)^{-\frac{1}{2}}+2d+\theta^{-1}d)+c_{5}\gamma d^{n}$

where $c_{3}=12\bar{c}(2\pi
e)^{\frac{n+1}{2}}(\bar{n}+2)^{\bar{n}+3}[(\bar{n}+2)!]^{-1}\big{(}\lvert\omega\rvert_{\mathcal{K}}+c\epsilon+1\big{)}\beta^{-\frac{\bar{n}+2}{\bar{n}+1}}\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}$,
$c_{5}=4\bar{\bar{c}}c_{4}m(\lvert\omega\rvert_{V}+c\epsilon+\max\limits_{1\leq
j\leq m}3|B^{0}_{j}|+1)$, $\bar{c}$ and $\bar{\bar{c}}$ depend only on $n$,
$\bar{n}$ and $\tau$, $c_{4}$ and $c$ are independent of $t$, $\epsilon$ and
$\gamma$. And for $\gamma$ small enough , $V_{\gamma}$ can be a set of
positive measures, and $meas(V\setminus V_{\gamma})\to 0$ as $\gamma\to 0$.

As for $\ k\in\mathbb{Z}^{n}\setminus\\{0\\}$, we have
$t\gamma_{v}|k|^{-\tau}\geq\frac{t\gamma_{v}}{({1+|k|)}^{\tau}}$, so for
$\xi\in V\setminus V_{*}$, (4.11) , (4.12) ,(4.13) and (4.14) are naturally
satisfied.

######  Remark 5.3.

As of $k=0$, we consider the small divisors which come from the coefficient
matrices. By (4.10), we just have to consider (4.8) for $k=0$. For fixed $j$,
by [5], we have

$\det(M^{v}_{k})=2\sec\theta^{t,v}_{j}e^{i\langle
k,t\omega_{v}\rangle}(\cos{\langle{k,t\omega_{v}}\rangle}-\cos{\theta^{t,v}_{j}})$

where $M^{v}_{k}$ represents the coefficient matrix of (4.8) in step $v$. When
$k=0$, $|\det(M^{v}_{0})|=2(|\sec\theta^{t,v}_{j}|-1)$. By the previous
definitions, $\underline{A^{v}_{t}}=I+t(A^{v}-I)$,
$\underline{A_{t,j}^{v}}=\sec\theta^{t,v}_{j}$ and
$A^{v}_{j}=\sec\theta^{v}_{j}$, then we have
$2(|\sec\theta^{t,v}_{j}|-1)=2t(|A^{v}_{j}|-1)=2t(|\sec\theta^{v}_{j}|-1)$. By
Lemma 4.3, we have $|B^{v}_{j}|\geq\frac{1}{2}|B^{0}_{j}|$ for $v=1,2,3\dots$,
where $B^{v}_{j}=\tan\theta^{v}_{j}$, $B^{0}_{j}$ is a constant. So for $t$
and $\gamma_{0}$ small enough, $v=1,2,3\dots$, we have

$|\det(M^{v}_{0})|=2t(|\sec\theta^{v}_{j}|-1)\geq
2t\\{((\frac{1}{2}|B^{0}_{j}|)^{2}+1)^{\frac{1}{2}}-1\\}\geq\frac{ct^{2}\gamma_{v}^{2}}{(|0|+1)^{2\tau}}$

which is consistent with $\ k\in\mathbb{Z}^{n}\setminus\\{0\\}$. And for
$v=0$, as $\theta_{j}^{0}$ is a constant, then
$|\det(M^{0}_{0})|=2t(|\sec\theta^{0}_{j}|-1)\geq\frac{ct^{2}\gamma_{0}^{2}}{(|0|+1)^{2\tau}}$
hold naturally for $t$ and $\gamma_{0}$ small enough. Since $j$ is arbitrary,
then we have (4.11) , (4.12) ,(4.13) and (4.14) are satisfied for $k=0$,
$\forall\ 1\leq j\leq m$ and $\forall\ v\in\mathbb{N}$. That is to say, for
$k=0$, the small divisor conditions hold naturally.

Thus, measure estimate in Theorem 5.3 is corresponding to all small divisor
conditions. The proof of Theorm 3.2 is complete. $\square$

Similar to [5], the proof of Theorem 3.1 is based on Theorem 3.2, and the
concrete proof refers to the proof of Theorem 1.1 in [5].

##  6 Applied to the symplectic difference scheme (1.4)

Let us focus on the symplectic difference scheme (1.4).

###  6.1 Proof of Corollary 3.1

Now, as for the symplectic difference scheme (1.4), we focus on Corollary 3.1
for $\epsilon$ and $t$ small enough. The case is simple, we just substitute
$\epsilon H_{1}+t^{s}P_{1}$ in (1.4) as a perturbation in (1), namely, let
$P^{{}^{\prime}}=\epsilon H_{1}+t^{s}P_{1}$ . Assume that
$\sup_{D}|P^{{}^{\prime}}|\leq
M_{1}\epsilon+2nM_{2}l_{*}t^{s}\leq\epsilon^{{}^{\prime}}$ , then by Theorem
3.2, we get the corresponding conclusions with $\epsilon^{{}^{\prime}}$ rather
than $\epsilon$, including the existence of invariant tori and measure
estimation. Then we get Corollary 3.1. $\square$

###  6.2 Proof of Corollary 3.2

As for the Corollary 3.2, its proof is obvious. $\square$

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