$J:=\left\\{\alpha=(\mathbf{q},\mathbf{p})\in{\mathbb{Z}}_{\mathbb{H}}^{m+n}:\|\mathbf{p}\|_{\infty}\leq
8m\|\mathbf{q}\|_{\infty}\right\\}$,

  2. ii.

$\beta:J\to{\mathbb{R}}_{+}$,
$\alpha=(\mathbf{q},\mathbf{p})\mapsto\beta_{\alpha}:=\max\left\\{\Phi_{1}^{-1}\left(|q_{1}|_{\infty}\right),\ldots,\Phi_{m}^{-1}\left(|q_{m}|_{\infty}\right)\right\\}$,

  3. iii.

$(u_{s})_{s\geq 1}$ given by $u_{s}=M^{s}$ for all $s\in{\mathbb{N}}$,

  4. iv.

For each $\alpha\in J$, write $R_{\alpha,t}:=\left\\{A_{j}\in
I_{\mathbb{H}}^{m\times 1}:\mathbf{q}\,A_{t}=p_{t}\right\\}$ for
$t\in\\{1,\ldots,n\\}$ and the resonant set $R_{\alpha}$ is
$R_{\alpha}:=\prod_{t=1}^{n}R_{\alpha,t}$,

  5. v.

In this context, $\kappa_{t}=1-\frac{1}{m}$ and $\delta_{t}=4m$ for
$t\in\\{1,\ldots,n\\}$.

Let $\rho:(\rho_{1},\ldots,\rho_{n}):{\mathbb{N}}\to{\mathbb{R}}_{+}^{n}$ be
given by

$\rho_{j}(u):=2\,M\frac{\varphi_{j}(u)}{\|\Phi(u)\|_{\infty}}\left(\prod_{s=1}^{n}\varphi_{s}(u)\prod_{k=1}^{m}\Phi_{k}(u)\right)^{-1/n}\qquad(1\leq
j\leq n,\;u\in{\mathbb{N}})$

and $\Psi=(\psi_{1},\ldots,\psi_{n}):{\mathbb{N}}\to{\mathbb{R}}_{+}^{n}$ by

$\psi_{j}(u):=\frac{\varphi_{j}(u)}{\|\Phi(u)\|_{\infty}}\quad(1\leq j\leq
n,\;u\in{\mathbb{N}}).$

Recall that, by Lemma 2, we can replace $\Psi$ with any of its multiples
without altering the measure. In view of Lemma 13, we may assume

$\prod_{t=1}^{n}\varphi_{t}(u)\prod_{r=1}^{m}\Phi_{r}(u)\leq\frac{1}{2^{m+n}}\quad(u\in{\mathbb{N}}).$

In order to prove Theorem 12, take $J$, $R_{\alpha}$ for $\alpha\in J$,
$\beta$, and $(u_{s})_{s\geq 1}$ as above, pick a sufficiently large $M$ and
put $l_{s}=M^{s-1}$ for all $s\in{\mathbb{N}}$. Given
$\textbf{a}=(a_{1},\ldots,a_{n})\in{\mathbb{R}}^{n}$ satisfying (5.14), define
$\tilde{\rho}=(\tilde{\rho}_{1},\ldots,\tilde{\rho}_{n}):{\mathbb{N}}\to{\mathbb{R}}^{n}$
by

$\tilde{\rho}_{j}(u)=\frac{2}{u^{a_{t}-1}}\quad(u\in{\mathbb{N}})$

for each $t=1,\ldots,N$.

######  Lemma 14.

The system $((R_{\alpha})_{\alpha\in J},\beta)$ is ubiquitous with respect to
$\rho$ and $(l_{s})_{s\geq 1}$, $(u_{s})_{s\geq 1}$. The same system is also
ubiquitous with respect to $\tilde{\rho}$ and $(\tilde{l}_{s})_{s\geq 1}$,
$(u_{s})_{s\geq 1}$ for a suitable $(\tilde{l}_{s})_{s\geq 1}$.

The proof of Lemma 14 follows closely the proofs of lemmas 7 and 10. We leave
the details to the reader.

###  6.2. Proof of Theorem 11

Condition (6.2) and the Cauchy condensation imply that the next series are
either both convergent or both divergent:

$\sum_{q=1}^{\infty}\frac{1}{q}\left(\prod_{t=1}^{n}\varphi_{t}(q)\prod_{r=1}^{m}\Phi_{r}(q)\right)^{4}\quad\text{
and
}\quad\sum_{s=0}^{\infty}\left(\prod_{t=1}^{n}\varphi_{t}(u_{s})\prod_{r=1}^{m}\Phi_{r}(u_{s})\right)^{4}.$

Hence, the divergence case is a consequence of Lemma 14 and Theorem 5. In
order to prove the convergence case, for each $u\in{\mathbb{N}}$, let
$\mathcal{A}^{\mathbb{H}}(u;\varphi,\Phi)$ be the collection of matrices $A\in
I_{\mathbb{H}}^{m\times n}$ such that for some non-zero
$\mathbf{q}\in{\mathbb{Z}}[i]^{1\times m}$ we have

$\displaystyle\llbracket\mathbf{q}A_{t}\rrbracket_{\infty}<\varphi_{t}(u),$
$\displaystyle\quad(1\leq t\leq n),$
$\displaystyle|q_{r}|_{\infty}\leq\Phi_{r}(u),$ $\displaystyle\quad(1\leq
r\leq m).$

Therefore,

$\mu_{m\times
n}^{\mathbb{H}}\left(\mathcal{A}^{\mathbb{H}}(u;\varphi,\Phi)\right)\ll_{m,n}\left(\prod_{t=1}^{n}\varphi_{t}(u)\prod_{r=1}^{m}\Phi_{r}(u)\right)^{4}$

and, as in the proof of Theorem 9,

$\limsup_{u\to\infty}\mathcal{A}^{\mathbb{H}}(u;\varphi,\Phi)\subseteq\limsup_{u\to\infty}\mathcal{A}^{\mathbb{H}}(u;\varphi,c_{2}\Phi).$

Finally, the Borel-Cantelli lemma gives the theorem.

###  6.3. Proof of Theorem 12

As in the complex setting, for any
$\boldsymbol{\tau}=(\tau_{1},\ldots,\tau_{n})\in{\mathbb{R}}^{n}$ such that
(5.2) holds, define the sets

$W_{\infty}^{\mathbb{H}}(\boldsymbol{\tau}):=\left\\{A\in
I_{\mathbb{H}}^{m\times
n}:\left\llbracket\mathbf{q}A_{t}\right\rrbracket_{\infty}<\frac{1}{\|\mathbf{q}\|_{\infty}^{\tau_{t}-1}}\quad(1\leq
t\leq n),\,\,\text{ for i. m.
}\mathbf{q}\in{\mathbb{Z}}_{\mathbb{H}}^{m}\right\\}$

and, for any $\gamma>0$,

$\widetilde{W}_{\infty}^{\mathbb{H}}(\gamma,\boldsymbol{\tau}):=\bigcap_{Q\in\tfrac{1}{2}{\mathbb{N}}}\bigcup_{\begin{subarray}{c}\mathbf{q}\in{\mathbb{Z}}_{\mathbb{H}}^{m}\\\
\|\mathbf{q}\|_{\infty}=Q\end{subarray}}\bigcup_{\begin{subarray}{c}\mathbf{p}\in{\mathbb{Z}}_{\mathbb{H}}^{n}\\\
\alpha=(\mathbf{q},\mathbf{p})\in
J\end{subarray}}\prod_{t=1}^{n}\Delta\left(R_{\alpha,t},\frac{\gamma}{\|\mathbf{q}\|_{\infty}^{\tau_{t}}}\right).$

The core of the argument is done under the assumption $\gamma=1$. For the
upper bound, note that

$\\#\left\\{\mathbf{q}\in{\mathbb{Z}}_{\mathbb{H}}^{m}:\|\mathbf{q}\|_{\infty}=Q\right\\}\asymp_{m}Q^{4m-1}\quad\left(Q\in\tfrac{1}{2}{\mathbb{N}}\right),$

Indeed, for any $\mathbf{q}\in{\mathbb{Z}}_{\mathbb{H}}$, the quaternion
$2\mathbf{q}$ is a Lipschitz integer and, for all
$Q\in\tfrac{1}{2}{\mathbb{N}}$, we have $\|\mathbf{q}\|_{\infty}=Q$ if and
only if $\|2\mathbf{q}\|_{\infty}=2Q$.

Also, for $t,l\in\\{1,\ldots,m\\}$, we need

$\asymp\left(\max\left\\{1,\frac{\|\mathbf{q}\|_{\infty}^{-\tau_{l}}}{\|\mathbf{q}\|_{\infty}^{-\tau_{t}}}\right\\}\|\mathbf{q}\|_{\infty}^{\tau_{j}(m-1)}\right)^{4}$

balls of radius $\|\mathbf{q}\|^{-\tau_{t}}$ to cover
$\triangle(R_{\alpha,l},\|\mathbf{q}\|_{\infty}^{-\tau_{l}})$. Hence, if
$s>0$, we have

$\displaystyle\mathcal{H}^{s}\left(\widetilde{W}^{\mathbb{H}}_{\infty}(1,\boldsymbol{\tau})\right)$
$\displaystyle\ll\liminf_{N\to\infty}\sum\limits_{Q\geq
N}Q^{4m-1}Q^{2n}Q^{-s\tau_{t}}Q^{4\tau_{t}n(m-1)}Q^{4\sum\limits_{\tau_{l}<\tau_{t}}\tau_{t}-\tau_{l}}$
$\displaystyle=\liminf_{N\to\infty}\sum\limits_{Q\geq
N}Q^{4m+4n-1+4\tau_{t}n(m-1)+4\sum\limits_{\tau_{l}<\tau_{j}}\tau_{t}-\tau_{l}}Q^{-s\tau_{t}}.$

The series converges if and only if $s>s_{t}(\boldsymbol{\tau})$.

From this point, the argument follows verbatim that of the complex case. That
is, we reorder the coefficients $\tau$ to have
$\tau_{1}\geq\ldots\geq\tau_{2}\geq\ldots\geq\tau_{n}>1$ and we consider two
cases: $\tau_{n}\geq\frac{n+m}{m}$ and $\frac{n+m}{m}>\tau_{n}$. Afterwards,
we apply Theorem 2 to conclude

$\dim_{H}\widetilde{W}^{\mathbb{H}}_{\infty}(1,\boldsymbol{\tau})=\min\left\\{s_{1}(\boldsymbol{\tau}),\ldots,s_{n}(\boldsymbol{\tau})\right\\}.$

The theorem for an arbitrary $\gamma$ follows from the continuity of the
dimension as a function of $\tau$. Finally, we conclude the theorem for
$W_{\infty}^{\mathbb{H}}(\boldsymbol{\tau})$ by appealing to the equivalence
of any two norms in a finite dimensional vector space.

##  7\. Formal power series approximation

In this section we study sets of linear forms over the field of formal power
series. Let ${\mathbb{F}}$ be the finite field with $t=p^{r}$ elements for
some prime $p$ and $r\in{\mathbb{N}}$. We define the field of Laurent series
with coefficients from $\mathbb{F}$ or the field of formal power series with
coefficients from $\mathbb{F}$ to be

(7.1)
$\mathcal{L}=\left\\{\sum\limits_{i=-n}^{\infty}a_{-i}X^{-i}\,:\,n\in{\mathbb{Z}},\
a_{i}\in{\mathbb{F}},\ a_{n}\neq 0\right\\}\cup\\{0\\}.$

An absolute value $\|\cdot\|$ on $\mathcal{L}$ can be defined as

$\left\|\sum\limits_{i=-n}^{\infty}a_{-i}X^{-i}\right\|=t^{n},\quad\,\left\|0\right\|=0.$

For any $\mathbf{x}=(x_{1},\ldots,x_{h})\in\mathcal{L}^{h}$, we define the
height of $\mathbf{x}$ to be

$\|\mathbf{x}\|_{\infty}=\max\\{\|x_{1}\|,\ldots,\|x_{h}\|\\}.$

Note that for both $\|\cdot\|$ and $\|\cdot\|_{\infty}$, we have

$\|x+y\|\leq\max(\|x\|,\|y\|)\quad\text{ and
}\quad\|\mathbf{x}+\mathbf{y}\|_{\infty}\leq\max(\|\mathbf{x}\|_{\infty},\|\mathbf{y}\|_{\infty}).$

In $\mathcal{L}$, the polynomial ring ${\mathbb{F}}[X]$ plays a role analogous
to the one played by the integers in the field of real numbers. We define the
polynomial part of a non-zero element by

$\left[\sum\limits_{i=-n}^{\infty}a_{-i}X^{-i}\right]=\sum\limits_{i=-n}^{0}a_{-i}X^{-i}.$

Define the distance to $\mathcal{L}[X]^{h}$ for a point
$\mathbf{x}\in\mathcal{L}^{h}$ as

$|\langle\mathbf{x}\rangle|=\min_{\mathbf{p}\in{\mathbb{F}}[X]^{h}}\|\mathbf{x}-\mathbf{p}\|_{\infty}\,.$

Let

$I_{\mathcal{L}}=\\{x\in\mathcal{L}:\,[x]=0\\}=B(0,1)=\\{x\in\mathcal{L}:\|x\|<1\\}.$

Fix $n,m\in{\mathbb{N}}$ and let $\mathcal{L}^{m\times n}$ to be the set of
$m\times n$ dimensional matrices with entries from $\mathcal{L}$ and
$I^{m\times n}_{\mathcal{L}}$ to be the $m\times n$ dimensional matrices with
entries from $I_{\mathcal{L}}$.
We will also make use of the fact about the number of polynomials of fixed
degree, namely

(7.2)
$\\#\\{\mathbf{q}\in{\mathbb{F}}[X]^{m}:\,\|\mathbf{q}\|_{\infty}=t^{r}\\}=m(r-1)t^{m-1}t^{tm}.$

Denote by $\mu^{\mathcal{L}}_{m\times n}$ the $mn$-dimensional Haar measure on
$\mathcal{L}^{m\times n}$ normalised by $\mu^{\mathcal{L}}_{m\times
n}(I^{m\times n}_{\mathcal{L}})=1$.

As in the previous sections, we have an $n$-tuple $\\{\phi_{i}\\}_{1\leq i\leq
n}$ and an $m$-tuple $\\{\Phi_{k}\\}_{1\leq k\leq m}$ of positive functions
defined on ${\mathbb{N}}$ such that

$\phi_{i}(u)\to 0\quad\text{ as }\quad u\to\infty\quad(1\leq i\leq n)$

and

$\Phi_{k}(u)\to\infty\quad\text{ as }\quad u\to\infty\quad(1\leq k\leq m).$

Define the set

$W^{\mathcal{L}}_{n,m}(\phi,\Phi):=\left\\{A\in I^{m\times
n}_{\mathcal{L}}:\begin{array}[]{l}\text{the system
}\left\\{\begin{array}[]{l}|\langle\mathbf{q}A_{i}\rangle|<\phi_{i}(u)\quad(1\leq
i\leq n),\\\\[8.61108pt] \|q_{k}\|\leq\Phi_{k}(u)\quad(1\leq k\leq
m),\end{array}\right.\\\\[12.91663pt] \text{ has a solution
}\mathbf{q}\in{\mathbb{F}}[X]^{m}\backslash\\{0\\}\text{ for i.m.
}u\in{\mathbb{N}}\end{array}\right\\}.$

We prove the following weighted version of the Khintchine-Groshve type theorem
for the formal power series.

######  Theorem 13.

Assume that there are constants $N_{0},M\in{\mathbb{N}}_{>1}$ and
$c_{1},c_{2}>1$ such that

(7.3) $c_{1}\Phi_{k}(M^{j})\leq\Phi_{k}(M^{j+1})\leq
c_{2}\Phi_{k}(M^{j}),\quad 1\leq k\leq m,\quad\forall j\in{\mathbb{N}}_{\geq
N_{0}}.$

Then

$\mu^{\mathcal{L}}_{m\times
n}(W^{\mathcal{L}}_{n,m}(\phi,\Phi))=\begin{cases}0\quad\text{\rm if
}\quad\sum\limits_{r=1}^{\infty}\frac{1}{r}\prod_{k=1}^{m}\Phi_{k}(r)\prod_{i=1}^{n}\phi_{i}(r)<\infty,\\\\[8.61108pt]
1\quad\text{\rm if
}\quad\sum\limits_{r=1}^{\infty}\frac{1}{r}\prod_{k=1}^{m}\Phi_{k}(r)\prod_{i=1}^{n}\phi_{i}(r)=\infty.\end{cases}$

Below we briefly highlight the results preceding this. These include but are
not limited to

  * •

$n=m=1$, $\psi$ monotonic, de Mathan [21].

  * •

$nm\geq 1$, $\Psi$ univariable monotonic, Kristensen [59]

  * •

$n\geq 1$, $m\geq 2$, $\Psi$ univariable monotonic inhomogeneuos, Kristensen
[61].

To sum up, nothing is known prior to our result in the weighted settings.
However, it is worth remarking that, an asymptotic formula for the number of
solutions to the Diophantine inequalities
$|\langle\mathbf{q}A\rangle|<\psi(\|\mathbf{q}\|)$ was proven in [25].

For the Hausdorff dimension result we consider a slightly different setup. For
any vector $\boldsymbol{\tau}=(\tau_{1},\ldots,\tau_{n})\in{\mathbb{R}}^{n}$,
satisfying

(7.4) $\min_{1\leq j\leq n}\tau_{j}>1\text{ \,\,\, and \,\,\,
}\sum_{j}\tau_{j}\geq n+m$

define the set

$W^{\mathcal{L}}_{n,m}(\tau):=\left\\{A\in I^{m\times
n}_{\mathcal{L}}:|\langle\mathbf{q}A_{j}\rangle|<\|\mathbf{q}\|_{\infty}^{-\tau_{j}}\|\mathbf{q}\|_{\infty}\quad(1\leq
j\leq n)\quad\text{ for i.m. }\mathbf{q}\in{\mathbb{F}}[X]^{m}\right\\}$

and quantities

$s_{j}(\tau):=n(m-1)+\,\frac{m+n-\sum\limits_{r\,:\,\tau_{r}<\tau_{j}}(\tau_{r}-\tau_{j})}{\tau_{j}}\;\quad(1\leq
j\leq n)\,.$

Then we have

######  Theorem 14.

For any vector $\boldsymbol{\tau}\in{\mathbb{R}}^{n}$, satisfying (7.4), we
have

$\operatorname{\dim_{H}}W^{\mathcal{L}}_{n,m}(\tau)=\min\\{s_{1}(\tau),\ldots,s_{n}(\tau)\\}.$

Previously known results regarding the Hausdorff dimension of this set
include:

  * •

$n=m=1$, Kristensen [59].

  * •

$nm\geq 1$, $\Psi$ non-monotonic, Kristensen [60].

Our weighted result is completely new.

###  7.1. Proof of Theorem 13

First note that the convergence case is a simple consequence of the Borel-
Cantelli lemma. The proof for the divergence case is similar to the complex
setup, so we will only highlight the differences. As in the other
applications, we will make use of the following form of Minkowski’s theorem on
linear forms.

######  Lemma 15 ([73]).

Suppose that for some $H\in{\mathbb{N}}$ one has

$\prod_{i=1}^{n}\phi_{i}(H)\prod_{j=1}^{m}\Phi_{j}(H)\geq t^{n+m}.$

Then for any $A\in I^{m\times n}_{\mathcal{L}}$, the system

$\displaystyle\|\mathbf{q}A_{i}-p_{i}\|<\phi_{i}(H)\hskip 25.6073pt(1\leq
i\leq n),$ $\displaystyle\|q_{j}\|\leq\Phi_{j}(H)\hskip 54.06006pt(1\leq j\leq
m)$

has a non-trivial solution $(\mathbf{q},\mathbf{p})\in{\mathbb{F}}[X]^{m+n}$.

We work with the following objects:

  1. i.

$J:=\\{\alpha=(\mathbf{q},\mathbf{p})=(q_{1},\dots,q_{m},p_{1},\ldots,p_{n})\in{\mathbb{F}}[X]^{m+n}\,:\,\|\mathbf{p}\|_{\infty}\leq\|\mathbf{q}\|_{\infty}\\}$,

  2. ii.

$\beta:J\to{\mathbb{R}}_{+},\,\alpha\mapsto\beta_{\alpha}=\max\left\\{\Phi_{1}^{-1}\left(\|q_{1}\|\right),\ldots,\Phi_{m}^{-1}\left(\|q_{m}\|\right)\right\\}$,

  3. iii.

$u_{k}=M^{k}$ for some $M\in{\mathbb{N}}_{\geq 2}$ to be chosen later,

  4. iv.

$J_{k}=\\{\alpha\in J:l_{k}\leq\beta_{\alpha}\leq u_{k}\\}$ for some suitable
sequence $l_{k}<u_{k}$ for all $k\in{\mathbb{N}}$.,

  5. v.

Resonant sets as $R_{\mathbf{q}}=\prod_{i=1}^{n}R_{\mathbf{q},i}$, where
$R_{\mathbf{q},i}=\left\\{A_{i}\in I^{m\times
1}_{\mathcal{L}}:\mathbf{q}A_{i}=p_{i}\text{ for some
}p_{i}\in{\mathbb{F}}[X]\right\\}$,

  6. vi.

In the current setting $\kappa_{i}=\frac{m-1}{m}$ and $\delta_{i}=m$ for each
$i=1,\dots,n$.

Finally, we define functions

$\rho_{j}(u)=\frac{\phi_{j}(u)}{\|\Phi(u)\|_{\infty}}\left(t^{-(m+n)}\prod_{s=1}^{n}\phi_{s}(u)\prod_{k=1}^{m}\Phi_{k}(u)\right)^{-1/n}$

and

$\psi_{j}(u)=\frac{\phi_{j}(u)}{\|\Phi(u)\|_{\infty}}.$

Also note that with these choices of functions we have

$\prod_{j=1}^{n}\frac{\psi_{j}(u)}{\rho_{j}(u)}=t^{m+n}\prod_{j=1}^{n}\phi_{j}(u)\prod_{k=1}^{m}\Phi_{k}(u)\quad(u\in{\mathbb{N}}).$

As in the other applications, using Minkowski’s theorem, we can make an
additional assumption on $\phi$ and $\Phi$. If there is a strictly increasing
sequence of natural numbers $(n_{j})_{j\geq 1}$ such that

$\prod_{s=1}^{n}\phi_{s}(n_{j})\prod_{k=1}^{m}\Phi_{k}(n_{j})\geq
t^{m+n}\quad(j\in{\mathbb{N}}),$

then $\mu^{\mathcal{L}}_{m\times n}(W^{\mathcal{L}}_{n,m}(\phi,\Phi))=1$ by
Lemma 15. Therefore, we will assume that

(7.5) $\prod_{j=1}^{n}\phi_{j}(u)\prod_{k=1}^{m}\Phi_{k}(u)<t^{m+n}\;\text{
for large }u\in{\mathbb{N}}.$

The main ingredient of the proof is the formal power series analogue of a
weighted ubiquitous system.

######  Lemma 16.

The system $(\left\\{R_{\alpha}\right\\}_{\alpha\in J},\beta)$ is a weighted
ubiquitous system with respect to the function $\rho$ given above.

For each $j\in\\{1,\ldots,n\\}$, let $f_{j}:{\mathbb{N}}\to{\mathbb{R}}_{+}$
be the function given by

$f_{j}(u):=\phi_{j}(u)\left(t^{-(m+n)}\prod_{s=1}^{n}\phi_{s}(u)\prod_{k=1}^{m}\Phi_{k}(u)\right)^{-1/n}\quad(u\in{\mathbb{N}}).$

Observe that $\prod_{j}f_{j}(u)\prod_{k}\Phi_{k}(u)=t^{m+n}$ for all
$u\in{\mathbb{N}}$.

######  Proposition 6.

For each $u\in{\mathbb{N}}$ and each $A\in I^{m\times n}_{\mathcal{L}}$ there
is some $\alpha=(\mathbf{q},\mathbf{p})\in J$ such that

$A\in\prod_{j=1}^{n}\Delta\left(\mathfrak{R}_{\alpha,j},\,\frac{f_{j}(u)}{\|\mathbf{q}\|_{\infty}}\right)\;\text{
and }\;\beta_{\alpha}\leq u.$

###### Proof.

By Lemma 15, for every $H\in{\mathbb{N}}$ and $A\in I^{m\times
n}_{\mathcal{L}}$ there exists
$(\mathbf{q},\mathbf{p})\in{\mathbb{F}}[X]^{m+n}\backslash\\{0\\}$, such that

$\displaystyle\|\mathbf{q}A_{i}-p_{i}\|<f_{i}(H)\hskip 28.45274pt(1\leq i\leq
n),$ $\displaystyle\|q_{j}\|\leq\Phi_{j}(H)\hskip 54.06006pt\,(1\leq j\leq
m).$

Then in each coordinate $i$ we have

$\|\mathbf{q}\|_{\infty}\operatorname{dist}_{\infty}(A_{i},R_{\mathbf{q},i})\leq\inf_{A_{i}^{\prime}\in
R_{\mathbf{q},i}}\|\mathbf{q}A_{i}-\mathbf{q}A_{i}^{\prime}\|\leq\|\mathbf{q}A_{i}-p_{i}\|<f_{i}(H).$

Dividing everything by $\|\mathbf{q}\|_{\infty}$ and noting that
$\|q_{k}\|\leq\Phi_{k}(u)$ for all $k$ finishes the proof. ∎

For each $j\in{\mathbb{N}}$, define

$\widetilde{J}_{j}:=\left\\{(\mathbf{q},\mathbf{p})\in{\mathbb{F}}[X]^{m+n}:\frac{\Phi_{k}(u_{j})}{M}\leq\|q_{k}\|\leq\Phi_{k}(u_{j})\quad(1\leq
k\leq m)\right\\}.$

If $\alpha=(\mathbf{q},\mathbf{p})\in\widetilde{J}_{j}$, then
$\beta_{\alpha}\leq u_{j}$. Since $\beta_{\alpha}\to\infty$ as
$\alpha\to\infty$, we may pick a sequence $l_{j}$ ensuring
$\widetilde{J}_{j}\subseteq J_{j}$. For each $j\in{\mathbb{N}}$ and
$k\in\\{1,\ldots,m\\}$, define the set

$J_{j,k}:=\left\\{\alpha\in
J:\left\|q_{k}\right\|\leq\frac{\Phi_{k}(u_{j})}{M}\quad\text{and}\quad\|q_{s}\|\leq\Phi_{s}(u_{j})\quad(s\in\\{1,\ldots
m\\}\setminus\\{k\\})\right\\}.$

Let $B=\prod_{k=1}^{n}B(X_{k};r)$ be an arbitrary ball. Then, for all
$s\in{\mathbb{N}}$,

$\displaystyle B$ $\displaystyle=B\cap\bigcup_{\alpha:\beta_{\alpha}\leq
u_{s}}\prod_{k=1}^{n}\Delta\left(R_{\alpha,k},\,\frac{f_{k}(u_{s})}{\|\mathbf{q}\|_{\infty}}\right)$
$\displaystyle=\left(B\cap\bigcup_{\alpha\in\widetilde{J}_{s}}\prod_{k=1}^{n}\Delta\left(R_{\alpha,k},\,\frac{f_{k}(u_{s})}{\|\mathbf{q}\|_{\infty}}\right)\right)\cup\left(B\cap\bigcup_{h=1}^{m}\bigcup_{\alpha\in
J_{s,h}}\prod_{k=1}^{n}\Delta\left(R_{\alpha,k},\,\frac{f_{k}(u_{s})}{\|\mathbf{q}\|_{\infty}}\right)\right).$

For any fixed $\mathbf{q}=(q_{1},\ldots,q_{m})$, the number of $p_{i}$ such
that the intersection is non-empty is not greater than
$4\|\mathbf{q}\|_{\infty}r$.

Finally, we can bound the Haar measure of the second term in the union from
above. This is an analogue of Proposition 5 from complex setup.

######  Proposition 7.

If $M\geq 2^{2n+1}t^{2m+n}m$, then for every large $s\in{\mathbb{N}}$ we have

$\displaystyle\mu^{\mathcal{L}}_{m\times
n}\left(B\cap\bigcup_{h=1}^{m}\bigcup_{\alpha\in
J_{s,h}}\prod_{j=1}^{n}\Delta\left(R_{\alpha,j},\,\frac{f_{j}(u_{s})}{\|\mathbf{q}\|_{\infty}}\right)\right)\leq\frac{1}{2}\mu^{\mathcal{L}}_{m\times
n}(B).$

###### Proof.

For each $s\in{\mathbb{N}}$ and $k\in\\{1,\ldots,m\\}$, write

$J_{s,k}^{\prime}:=\left\\{\mathbf{q}\in{\mathbb{F}}[X]^{m}:\alpha=(\mathbf{q},\mathbf{p})\in
J_{s,k}\right\\},$

so we can bound the number of elements as

$\\#J_{s,k}^{\prime}\leq\frac{t^{m}}{M}\prod_{j=1}^{m}\Phi_{j}(u_{s}).$

If $G_{\mathcal{L}}$ is the intersection in the statement of Proposition 7,
then

$\displaystyle\mu^{\mathcal{L}}_{m\times n}(G_{\mathcal{L}})$
$\displaystyle\leq\sum\limits_{k=1}^{m}\sum\limits_{\mathbf{q}\in
J_{s,k}^{\prime}}(4r\|\mathbf{q}\|_{\infty})^{n}\prod_{j=1}^{n}\,\frac{f_{j}(u_{s})r^{m-1}}{\|\mathbf{q}\|_{\infty}}$
$\displaystyle\leq 4^{n}\sum\limits_{k=1}^{m}\sum\limits_{\mathbf{q}\in
J_{s,k}^{\prime}}r^{nm}\prod_{j=1}^{n}\,f_{j}(u_{s})$
$\displaystyle\leq\frac{4^{n}r^{nm}t^{m+n}}{\Phi_{1}(u_{s})\ldots\Phi_{m}(u_{s})}\sum\limits_{k=1}^{m}\\#J_{s,k}^{\prime}$
$\displaystyle\leq\frac{4^{n}t^{2m+n}m}{M}r^{nm}$
$\displaystyle\leq\frac{1}{2}\mu^{\mathcal{L}}_{m\times n}(B).$

∎

The definition of $\widetilde{J}_{s}$ tells us that for each
$\alpha=(\mathbf{q},\mathbf{p})\in\widetilde{J}_{s}$ one has

$\|\mathbf{q}\|_{\infty}\geq\frac{1}{M}\left\|\Phi_{1}(u_{s}),\ldots,\Phi_{m}(u_{s})\right\|_{\infty},$

so

$\displaystyle\mu^{\mathcal{L}}_{m\times n}\left(B\cap\bigcup_{\alpha\in
J_{s}}\prod_{k=1}^{m}\Delta\left(R_{\alpha,k},M\,\frac{f_{k}(u_{s})}{\|\Phi(u_{s})\|_{\infty}}\right)\right)$
$\displaystyle\geq\mu^{\mathcal{L}}_{m\times
n}\left(B\cap\bigcup_{\alpha\in\widetilde{J}_{n}}\prod_{k=1}^{m}\Delta\left(R_{\alpha,k},M\,\frac{f_{k}(u_{s})}{\|\Phi(u_{s})\|_{\infty}}\right)\right)$
$\displaystyle\geq\mu^{\mathcal{L}}_{m\times
n}\left(B\cap\bigcup_{\alpha\in\widetilde{J}_{s}}\prod_{k=1}^{m}\Delta\left(R_{\alpha,k},\,\frac{f_{k}(u_{s})}{\|\mathbf{q}\|_{\infty}}\right)\right)$
$\displaystyle\geq\frac{1}{2}\mu^{\mathcal{L}}_{m\times n}(B).$

So the ubiquity property is proven.

The rest of the proof of Theorem 13 is very similar to the complex setup, but
with the difference that we are considering

$\prod_{j=1}^{n}\varphi_{j}(q)\prod_{k=1}^{m}\Phi_{k}(q)$

instead of

$\left(\prod_{j=1}^{n}\varphi_{j}(q)\prod_{k=1}^{m}\Phi_{k}(q)\right)^{2}$

and therefore we skip it.

###  7.2. Proof of Theorem 14

####  7.2.1. Upper bound.

We make a natural cover of the limsup set and then use the Hausdorff-Cantelli
lemma.

$W^{\mathcal{L}}_{n,m}(\tau)\subseteq\bigcap_{Q\in{\mathbb{N}}}\bigcup_{\begin{subarray}{c}\mathbf{q}\in{\mathbb{F}}[X]^{m}\\\
\|\mathbf{q}\|_{\infty}=t^{Q}\end{subarray}}\bigcup_{\begin{subarray}{c}\mathbf{p}:\\\
(\mathbf{q},\mathbf{p})\in
J\end{subarray}}\prod_{j=1}^{n}\triangle\left(R_{\alpha,j},\|\mathbf{q}\|_{\infty}^{-\tau_{j}}\right).$

Also, observe that for all $j,l\in\\{1,\ldots,n\\}$ the number of balls of
radius $\|\mathbf{q}\|_{\infty}^{-\tau_{j}}$ to cover
$\triangle(R_{\alpha,l},\|\mathbf{q}\|_{\infty}^{-\tau_{l}})$ is
asymptotically equivalent to

$\max\left\\{1,\frac{\|\mathbf{q}\|_{\infty}^{-\tau_{l}}}{\|\mathbf{q}\|_{\infty}^{-\tau_{j}}}\right\\}\|\mathbf{q}\|_{\infty}^{\tau_{j}(m-1)}.$

Hence, for all $s>0$,

$\displaystyle\mathcal{H}^{s}\left(W(\tau)\right)$
$\displaystyle\ll\liminf_{N\to\infty}\sum\limits_{\begin{subarray}{c}\mathbf{q}\in{\mathbb{F}}[X]^{m}\\\
\|\mathbf{q}\|_{\infty}\geq
t^{N}\end{subarray}}\|\mathbf{q}\|_{\infty}^{m-1}\|\mathbf{q}\|_{\infty}^{n}\|\mathbf{q}\|_{\infty}^{-s\tau_{j}}\|\mathbf{q}\|_{\infty}^{\tau_{j}n(m-1)}\|\mathbf{q}\|_{\infty}^{\sum\limits_{l:\,\tau_{j}>\tau_{l}}(\tau_{j}-\tau_{l})}$
$\displaystyle=\liminf_{N\to\infty}\sum\limits_{\begin{subarray}{c}\mathbf{q}\in{\mathbb{F}}[X]^{m}\\\
\|\mathbf{q}\|_{\infty}\geq
t^{N}\end{subarray}}\|\mathbf{q}\|_{\infty}^{m+n-1+\tau_{j}n(m-1)+\sum\limits_{l:\,\tau_{j}>\tau_{l}}(\tau_{j}-\tau_{l})}\|\mathbf{q}\|_{\infty}^{-s\tau_{j}}$
$\displaystyle=m(t-1)t^{m-1}\liminf_{N\to\infty}\sum\limits_{r=N}^{\infty}\left(t^{m+n-1+\tau_{j}n(m-1)+\sum\limits_{l:\,\tau_{j}>\tau_{l}}(\tau_{j}-\tau_{l})-s\tau_{j}}\right)^{r}.$

The series above converges if and only if the exponent of $e$ is strictly less
than $0$, or equivalently, $s>s_{j}(\tau)$. The result follows by taking the
infimum over $j$.

####  7.2.2. Lower bound.

The proof of the lower bound is similar to the complex case with obvious
modifications. Namely, given any
$\boldsymbol{l}=(l_{1},\ldots,l_{n})\in{\mathbb{R}}^{n}$ such that

(7.6) $\min_{1\leq j\leq n}l_{j}\geq 1\;\text{ and
}\;\sum\limits_{j=1}^{n}l_{j}=m+n,$

we define
$\tilde{\rho}=(\tilde{\rho}_{1},\ldots,\tilde{\rho}_{n}):{\mathbb{N}}\to{\mathbb{R}}^{n}$
by

$\tilde{\rho}_{j}(u)=\,\frac{1}{u^{l_{j}-1}}\quad(1\leq j\leq
n,\;u\in{\mathbb{N}}).$

and then prove the ubiquity statement for this set of functions as in Lemma
16. For the last part of the proof, we reorder the coefficients $\tau$ to have
$\tau_{1}\geq\ldots\geq\tau_{2}\geq\ldots\geq\tau_{n}>1$ and then we consider
two cases: $\tau_{n}\geq\frac{n+m}{m}$ and $\frac{n+m}{m}>\tau_{n}$ and then
apply Theorem 2 to get the statement of the theorem. We refer reader to the
Section 4.3, as the choice of parameters and calculations are the same as in
the $p$-adic case.

##  8\. Uniformly distributed sequences

Fix a natural number $n$. Let $\omega=(\boldsymbol{\omega}_{j})_{j\geq 1}$ be
a sequence of points in $[0,1]^{n}$ (that is,
$\boldsymbol{\omega}_{j}=(\omega_{j,1},\dots,\omega_{j,n})\in[0,1]^{n}$ for
each $j\in{\mathbb{N}}$) and let $\psi_{i}:{\mathbb{N}}\to{\mathbb{R}}_{+}$ be
a monotonic decreasing function for each $1\leq i\leq n$. Define
$\Psi=(\psi_{1},\ldots,\psi_{n})$. Consider the set

$W_{\omega}(\Psi):=\left\\{\mathbf{x}=(x_{1},\ldots,x_{n})\in[0,1]^{n}:|x_{i}-\omega_{j,i}|<\psi_{i}(j)\,\quad(1\leq
i\leq n)\quad\text{ for i.m. }j\in{\mathbb{N}}\right\\}\,.$

By a rectangle $R$ in $[0,1]^{n}$ we mean a set of the form

$R=\left\\{\mathbf{x}\in[0,1]^{n}:a_{i}\leq x_{i}<b_{i}\quad(1\leq i\leq
n)\,\right\\}$

for some $(a_{1},\dots,a_{n}),(b_{1},\dots,b_{n})\in[0,1]^{n}$ with
$a_{i}<b_{i}$ for each $i=1,\ldots,n$. For any rectangle $R$ and any
$N\in{\mathbb{N}}$, define

$A(R;N,\omega):=\\#\\{1\leq j\leq N:\boldsymbol{\omega}_{j}\in R\\}\,.$

Recall that $\mu_{n}^{{\mathbb{R}}}$ is the Lebesgue measure on $[0,1]^{n}$.

######  Definition 3 (Uniformly distributed sequence).

A sequence $\omega=(\boldsymbol{\omega}_{j})_{j\geq 1}$ of points in
$[0,1]^{n}$ is a uniformly distributed sequence on $[0,1]^{n}$, denoted
$\omega$ is a $u.d.s$ on $[0,1]^{n}$, if for any $n$-dimensional rectangle
$R\subseteq[0,1]^{n}$ we have

$\lim_{N\to\infty}\frac{A(R;N,\omega)}{N}=\mu_{n}^{{\mathbb{R}}}(R)\,.$

Much is known when $\omega$ is a sequence of independent identically
distributed uniform random variables. Initiated by Fan and Wu [32], who
considered the one dimensional case, the topic has since been investigated and
generalised by numerous authors, see for example [28, 29, 30, 34]. These
results strongly depend on the randomness of the sequence $\omega$. As a
classical example in Diophantine approximation of a uniformly distributed
sequence that is certainly not random consider the sequence
$\omega=(\\{n\alpha\\})_{n\in{\mathbb{N}}}$ for some $\alpha\in{\mathbb{R}}$,
where $\\{x\\}$ denotes the fractional part of $x\in{\mathbb{R}}$. The metric
properties of $W_{(\\{n\alpha\\})_{n\in{\mathbb{N}}}}(\Psi)$ in one dimension
have been studied in [16, 33, 36, 53, 54, 63, 75] and in higher dimensions in
[8, 44, 55, 68, 72]. In this setting, it was shown by Chebyshev (in one
dimension) and by Khintchine (in higher dimensions) that for
$\alpha\in{\mathbb{R}}\backslash{\mathbb{Q}}$ the sequence
$(\\{n\alpha\\})_{n\in{\mathbb{N}}}$ is well distributed [51]. Recently,
Moshchevitin proved metric properties on well distributed sequences [67] (his
definition of well distributed sequence differs from the classic one). In the
following section we begin with a general statement in which we consider no
other conditions on $\omega$ other than being uniformly distributed.

In our previous applications of weighted ubiquity, we used an adequate version
of Minkowski’s theorem for linear forms to ensure that certain neighbourhoods
of the resonant sets intersect a large portion of any ball. In our current
setting, we replace Minkowski’s theorem with a condition on the discrepancy of
$\omega$.

######  Definition 4 (Discrepancy, Star-discrepancy).

Let $\omega$ be a sequence of points in $[0,1]^{n}$.

  1. i.

The discrepancy of $\omega$ is the function
$D:{\mathbb{N}}\to{\mathbb{R}}_{+}$, $N\mapsto D_{N}(\omega)$, given by

$D_{N}=D_{N}(\omega):=\sup_{R\subseteq[0,1]^{n}}\left|\frac{A(R;N,\omega)}{N}-\mu_{n}^{{\mathbb{R}}}(R)\right|,$

where the supremum is taken over all rectangles $R\subseteq[0,1]^{n}$.

  2. ii.

The star-discrepancy of $\omega$ is the function
$D^{*}:{\mathbb{N}}\to{\mathbb{R}}_{+}$, $N\mapsto D_{N}^{*}(\omega)$, given
by

$D_{N}^{*}=D_{N}^{*}(\omega):=\sup_{R\subseteq[0,1]^{n}}\left|\frac{A(R;N,\omega)}{N}-\mu_{n}^{{\mathbb{R}}}(R)\right|,$

where the supremum is taken over all rectangles $R\subseteq[0,1]^{n}$ of the
form $[0,t_{1})\times\cdots[0,t_{n})$ for some $t_{1},\ldots,t_{n}>0$.

It is well known [62, Chapter 2] that for any sequence $\omega$ in $[0,1]^{n}$
and any $N\in{\mathbb{N}}$ we have

$D_{N}^{*}(\omega)\leq D_{N}(\omega)\leq 2^{n}D_{N}^{*}(\omega).$

######  Definition 5 ($({\mathcal{N}},v)$-Discrepancy satisfying sequence).

Let $\omega=(\boldsymbol{\omega}_{j})_{j\geq 1}$ be a sequence of points in
$[0,1]^{n}$. We say that $\omega$ is a $({\mathcal{N}},v)$-Discrepancy
satisfying sequence, which we write as $({\mathcal{N}},v)$-d.s.s, if there
exist a sequence ${\mathcal{N}}=(N_{i})_{i\geq 1}$ of strictly increasing
integers and a monotonic decreasing function
$v:{\mathbb{N}}\to{\mathbb{R}}_{+}$ with $v(N)\to 0$ as $N\to\infty$ such that

$D_{N_{i}}(\omega)<v(N_{i})\quad\text{ and }\quad
N_{i-1}v(N_{i})<2^{-(n+3)}\quad(i\in{\mathbb{N}}).$

The constant $2^{-(n+3)}$ can be modified by adapting the computations
accordingly. Clearly, every uniformly distributed sequence is a
$(\mathcal{N},v)$-d.s.s. for some $\mathcal{N}$ and $v$. This follows from the
observation that $\omega$ is a u.d.s if and only if the discrepancy tends to
zero [62, Chapter 2, Example 1.2]. Hence one can choose an increasingly sparse
sequence ${\mathcal{N}}$ satisfying the conditions of Definition 5.

Under this setting, we prove the following Khintchine-type.

######  Theorem 15.

Let $\Psi=(\psi_{1},\dots,\psi_{n})$ be an n-tuple of monotonic decreasing
functions. Suppose that $(\boldsymbol{\omega}_{i})_{i\geq 1}$ is a
$({\mathcal{N}},v)$-d.s.s. for ${\mathcal{N}}=(N_{i})_{i\geq 1}$ and
$v:{\mathbb{N}}\to{\mathbb{R}}_{+}$, that $\Psi$ is $c$-regular for
${\mathcal{N}}$ and that for some vector
$\boldsymbol{\tau}=(\tau_{1},\ldots,\tau_{n})\in{\mathbb{R}}_{+}$ verifying

$\min\\{\tau_{1},\ldots,\tau_{n}\\}>0,\quad\sum_{j=1}^{n}\tau_{j}=1$

every large $N\in{\mathbb{N}}$ satisfies

$\psi_{j}(N)\leq D_{N}^{\tau_{j}}\quad(1\leq j\leq n).$

Then

$\mu_{n}^{{\mathbb{R}}}(W_{\omega}(\Psi))=1\quad\text{\rm if
}\quad\sum_{j=1}^{\infty}v(N_{j})^{-1}\prod_{i=1}^{n}\psi_{i}(N_{j})=\infty.$

The regularity condition imposed on the approximation functions in Theorem 15
is rather restrictive. It might be possible to relax it but, we believe, it
might not be possible within the methods used in this paper.

In [52], Kiefer showed that for almost every sequence $\omega$ we have

(8.1) $\limsup_{N\to\infty}\;D_{N}^{*}(\omega)\,\sqrt{\frac{2N}{\log\log
N}}=1.$

Here and in the following, when we talk about properties holding for almost
every sequence in $[0,1]^{n}$ we have in mind the probability space of
sequences in $[0,1]^{n}$ along with its Borel $\sigma$-algebra and the product
measure $\mu_{\infty}$ induced by the Lebesgue measure on each factor (see
[47, Theorem 8.23]). The discrepancy of a sequence cannot converge too fast to
zero. A uniform lower bound on the discrepancy was obtained by K. Roth (see
[62, Chapter 2, Theorem 2.1]): if $n\in{\mathbb{N}}_{\geq 2}$, then every
sequence $\omega$ in $[0,1]^{n}$ verifies

(8.2) $D_{N}(\omega)\gg_{n}\frac{(\log N)^{\frac{n-1}{2}}}{N}.$

Moreover, by a theorem of van Aardenne-Ehrenfest [62, Chapter 2, Corollary
2.1], we have

$\limsup_{N\to\infty}\,ND_{N}^{*}(\omega)=\infty.$

######  Theorem 16.

Let $(\tau_{1},\ldots,\tau_{n})\in{\mathbb{R}}_{+}$ satisfy

$\min\\{\tau_{1},\ldots,\tau_{n}\\}>0\quad\text{ and
}\quad\sum_{j=1}^{n}\tau_{j}=1.$

Consider $\Psi=(\psi_{1},\dots,\psi_{n})$ such that for every large
$N\in{\mathbb{N}}$

$\psi_{j}(N)\leq\left(\frac{\log\log
N}{2N}\right)^{\frac{\tau_{j}}{2}}\quad(1\leq j\leq n).$

For almost every uniformly distributed sequence $\omega$, if there is some
$M>1$ such that

$\sum_{j=1}^{\infty}\frac{M^{\frac{3^{j}}{2}}}{\sqrt{j}}\prod_{i=1}^{n}\psi_{i}\left(M^{3^{j}}\right)=\infty,$

then $\mu_{n}^{{\mathbb{R}}}\left(W_{\omega}(\Psi)\right)=1$.

######  Definition 6 (Low discrepancy).

A sequence $\omega$ has low discrepancy if

$D_{N}(\omega)\ll\frac{(\log N)^{n}}{N}.$

The sequence $(M^{3^{j}})_{j\geq 1}$ can be replaced with $(M^{j^{2}})_{j\geq
1}$ when $\omega$ is a low-discrepancy sequence. For modern aspects of low
discrepancy sequences, see [65, 78] and the references therein.

######  Theorem 17.

Let $(\tau_{1},\ldots,\tau_{n})\in{\mathbb{R}}_{+}$ satisfy

$\min\\{\tau_{1},\ldots,\tau_{n}\\}>0\quad\text{ and
}\quad\sum_{j=1}^{n}\tau_{j}=1.$

Let $\omega$ be a low-discrepancy sequence and consider
$\Psi=(\psi_{1},\dots,\psi_{n})$ such that for every large $N\in{\mathbb{N}}$

$\psi_{j}(N)\leq D_{N}^{\tau_{j}}\quad(1\leq j\leq n).$

If for some $M>1$ we have

$\sum_{j=1}^{\infty}\frac{M^{j^{2}}}{j^{2n}}\prod_{i=1}^{n}\psi_{i}\left(M^{j^{2}}\right)=\infty,$

then $\mu_{n}^{{\mathbb{R}}}\left(W_{\omega}(\Psi)\right)=1$.

A similar problem was addressed by Boshernitzan and Chaika in the non-weighted
case [14, Example 8]. As they point out, part of the proof Theorem 18 may be
obtained from the work of Beresnevich, Dickinson, and Velani on ubiquitous
systems [9].

######  Theorem 18 ([14, Example 1]).

Take $n\in{\mathbb{N}}$. Let $\psi:{\mathbb{N}}\to{\mathbb{R}}_{+}$ be any
positive function and define the $n$-tuple $\Psi:=(\psi,\ldots,\psi)$. Almost
every sequence $\omega=(\boldsymbol{\omega}_{j})_{j\geq 1}$ on $[0,1]^{n}$
satisfies

$\mu_{n}^{{\mathbb{R}}}\left(W_{\omega}(\Psi)\right)=\begin{cases}0\quad\text{\rm
if }\quad\displaystyle\sum_{j=1}^{\infty}\psi^{n}(j)<\infty,\\\
1\quad\text{\rm if
}\quad\displaystyle\sum_{j=1}^{\infty}\psi^{n}(j)=\infty.\end{cases}$

There are examples of uniformly distributed sequences
$\omega=(\boldsymbol{\omega}_{j})_{j\geq 1}$ for which there is some positive
function $\psi:{\mathbb{N}}\to{\mathbb{R}}_{+}$ satisfying

$\sum_{j=1}^{\infty}\psi^{n}(j)=\infty\quad\text{ and
}\quad\mu_{n}^{{\mathbb{R}}}\left(W_{\omega}(\psi,\ldots,\psi)\right)<1.$

In fact, this is the case when $n=1$, $\alpha$ is an irrational number which
is not badly approximable, and $\omega_{j}=\\{j\alpha\\}$ for
$j\in{\mathbb{N}}$ [14, Remark 6].

We believe that the weighted version of Theorem 18 is true.

######  Conjecture 1.

For $\mu_{\infty}$-almost every sequence $\omega$ in $[0,1]^{n}$, for every
$n$-tuple of non-increasing functions $\Psi=(\psi_{1},\ldots,\psi_{n})$ we
have

$\mu_{n}^{{\mathbb{R}}}(W_{\omega}(\Psi))=\begin{cases}0\quad\text{\rm if
}\quad\displaystyle\sum_{j=1}^{\infty}\prod_{i=1}^{n}\psi_{i}(j)<\infty,\\\
1\quad\text{\rm if
}\quad\displaystyle\sum_{j=1}^{\infty}\prod_{i=1}^{n}\psi_{i}(j)=\infty.\\\
\end{cases}$

The convergence half is a consequence of the Borel-Cantelli lemma. The
divergence part is expected in view of Theorem 18 and since we can interpret a
uniformly distributed sequence as a generic realization of a sequence of
independent identically distributed uniform random variables.

We compute the Hausdorff dimension of $W_{\omega}(\Psi)$ for certain functions
$\Psi$ and sequences $\omega$ with low discrepancy.

######  Theorem 19.

Let $\omega$ be a low discrepancy sequence and let
$\boldsymbol{\tau}=(\tau_{1},\dots,\tau_{n})\in{\mathbb{R}}^{n}_{+}$ satisfy

$\sum_{i=1}^{n}\tau_{i}>1\,.$

Then, for $\Psi(N)=(N^{-\tau_{1}},\dots,N^{-\tau_{n}})$, we have that

$\operatorname{\dim_{H}}W_{\omega}(\Psi)=\min_{1\leq j\leq
n}\left\\{\frac{1+\sum_{i:\tau_{j}\geq\tau_{i}}(\tau_{j}-\tau_{i})}{\tau_{j}}\right\\}.$

We have the following result for exceptional uniformly distributed sequences.

######  Theorem 20.

Let $n=1$ and $\omega$ be a uniformly distributed sequence such that

$D_{N}(\omega)\ll\frac{1}{N}$

for infinitely many $N\in{\mathbb{N}}$ with the implied constant independent
of $N$. Let $\tau>1$. Then, for $\Psi(N)=N^{-\tau}$, we have that

$\operatorname{\dim_{H}}W_{\omega}(\Psi)=\frac{1}{\tau}.$

The result of Theorem 20 holds in higher dimensions but, as stated earlier,
for $n\geq 2$ Roth’s Theorem implies that

$D_{N}(\omega)\gg\frac{(\log N)^{\tfrac{n-1}{2}}}{N}\,,$

for all large $N\in{\mathbb{N}}$.

From this theorem one can deduce, in combination with the Three distance
theorem and a result of Khintchine [48, Satz 1] (that only ${\mathbb{Q}}$ is
singular in one dimension), the following theorem of Bugeaud [16, Theorem 1].

######  Corollary (Bugeaud [16]).

Let $\alpha\in{\mathbb{R}}\backslash{\mathbb{Q}}$. Then, for any $\tau>1$,

$\operatorname{\dim_{H}}W_{(\\{n\alpha\\})_{n\geq 1}}(\Psi)=\frac{1}{\tau}\,.$

Our results on uniformly distributed sequences are far from being optimal. A
refinement of our strategy may certainly improve our theorems. However, when
$n\geq 2$, Roth’s lower bound on the discrepancy (8.2) and our condition on
the $(\mathcal{N},v)$-discrepancy imply that the growth of
$\mathcal{N}=(N_{j})_{j\geq 1}$ is faster than exponential. As a consequence,
if we are after a 0-1 dichotomy, we will have to impose some conditions on the
approximation functions. A more promising landscape appears when we replace
the discrepancy of the sequence $\omega$ with its dispersion [27, Section
1.1.3.].

###  8.1. The ubiquity property

In this subsection we formulate a key statement which will allow us to
construct ubiquitous systems.

######  Proposition 8.

Let $\omega$ be a $({\mathcal{N}},v)$-Discrepancy satisfying sequence. Let
$\rho=(\rho_{1},\dots,\rho_{n})$ be an $n$-tuple of functions
$\rho_{i}:{\mathcal{N}}\to{\mathbb{R}}_{+}$ such that

  1. i)

each $\rho_{i}(N)\to 0$ as $N\to\infty$ ,

  2. ii)

$\prod_{i=1}^{n}\rho_{i}(N)=v(N)$ for all $N\in{\mathbb{N}}$ .

Then for any ball $B\subset[0,1]^{n}$ there exists sufficiently large
$k_{0}\in{\mathbb{N}}$ such that for all $k\geq k_{0}$

$\mu_{n}^{{\mathbb{R}}}\left(B\cap\bigcup_{N_{k-1}<j\leq
N_{k}}\Delta\left(\omega_{j},\rho(N_{k})\right)\right)\geq\frac{1}{4}\mu_{n}^{{\mathbb{R}}}(B).$

###### Proof.

Proposition 8 follows from combining Lemma 17 and Lemma 18 below. The first of
which, for our purposes, acts as a uniformly distributed version of
Minkowski’s Theorem for systems of linear forms.

######  Lemma 17.

Let $\rho=(\rho_{1},\dots,\rho_{n})$ be an $n$-tuple of functions
$\rho_{i}:{\mathbb{N}}\to{\mathbb{R}}_{+}$ with each $\rho_{i}(N)\to 0$ as
$N\to\infty$, and suppose that

$\prod_{i=1}^{n}\rho(N)=v(N)\quad(N\in{\mathbb{N}}).$

Then, for any ball $B\subset[0,1]^{n}$ and all sufficiently large
$k\in{\mathbb{N}}$

$\mu_{n}^{{\mathbb{R}}}\left(B\cap\bigcup_{1\leq j\leq
N_{k}}\Delta\left(\omega_{j},\rho(N_{k})\right)\right)\geq\frac{1}{2}\mu_{n}^{{\mathbb{R}}}(B).$

###### Proof.

Since $\omega$ is uniformly distributed on $[0,1]^{n}$ for any $\varepsilon>0$
there exists $k_{\varepsilon}\in{\mathbb{N}}$ such that for all
$k>k_{\varepsilon}$

(8.3) $A_{N_{k}}(\omega,B):=\\#\left\\{1\leq j\leq N_{k}:\omega_{j}\in
B\right\\}\geq N_{k}(\mu_{n}^{{\mathbb{R}}}(B)-\varepsilon).$

From now on, assume $k>k_{0}$ is such that

$\eqref{uniformly distributed used}\text{ holds for all
$\varepsilon<\tfrac{\mu_{n}^{{\mathbb{R}}}(B)}{2}$ and }\max_{1\leq i\leq
n}\rho_{i}(N_{k})<\frac{1}{2}r(B)\,.$

Then $\varepsilon>0$ is small enough such that

$A_{N_{k}}(\omega,B)\geq\frac{N_{k}}{2}\mu_{n}^{{\mathbb{R}}}(B).$

Now cover $\frac{1}{2}B$ by disjoint rectangles
$\Delta(x_{j},\tfrac{1}{2}\rho(N_{k}))$ for some finite set $\\{x_{j}\\}_{j}$
such that each $x_{j}\in\frac{1}{2}B$. Observe that

(8.4) $\
\bigcup_{j}\Delta\left(x_{j},\tfrac{1}{2}\rho(N_{k})\right)\supseteq\frac{1}{2}B\,.$

Note

$\mu_{n}^{{\mathbb{R}}}\left(\Delta(x_{j},\tfrac{1}{2}\rho(N_{k})\right)=\prod_{i=1}^{n}\rho_{i}(N_{k})=v(N_{k}).$

Since $\omega$ is a Dirichlet $({\mathcal{N}},v)$-Discrepancy satisfying
sequence note that

$\left|\frac{A_{N_{k}}\left(\omega,\Delta(x_{j},\tfrac{1}{2}\rho(N_{k}))\right)}{N_{k}}-\mu_{n}^{{\mathbb{R}}}\left(\Delta(x_{j},\tfrac{1}{2}\rho(N_{k}))\right)\right|<v(N_{k}),$

hence $A_{N_{k}}\left(\omega,\Delta(x_{j},\tfrac{1}{2}\rho(N_{k}))\right)\geq
1$ for each $x_{j}$. Thus to each $x_{j}$ we can associate a
$\omega_{\ell(j)}$ for some $1\leq\ell(j)\leq N_{k}$. Write
$x_{j}\sim\omega_{\ell(j)}$. Then for each $x_{j}\sim\omega_{\ell(j)}$

(8.5)
$\Delta\left(\omega_{\ell(j)},\rho(N_{k})\right)\supset\Delta\left(x_{j},\tfrac{1}{2}\rho(N_{k})\right),$

and so

$\mu_{n}^{{\mathbb{R}}}\left(B\cap\bigcup_{1\leq j\leq
N_{k}}\Delta(\omega_{j},\rho(N_{k}))\right)\overset{\eqref{uniform cover
2}}{\geq}\mu_{n}^{{\mathbb{R}}}\left(B\cap\bigcup_{1\leq j\leq
N_{k}}\bigcup_{x_{j}\sim\omega_{\ell(j)}}\Delta(x_{j},\frac{1}{2}\rho(N_{k}))\right)\overset{\eqref{uniform
cover 1}}{\geq}\frac{1}{2}\mu_{n}^{{\mathbb{R}}}\left(B\right)\,.$

∎

######  Lemma 18.

For any $B\subset[0,1]^{n}$ and all sufficiently large $k\in{\mathbb{N}}$
(with the size of $k$ dependent only on $B$).

$\mu_{n}^{{\mathbb{R}}}\left(B\cap\bigcup_{1\leq j\leq
N_{k-1}}\Delta(\omega_{j},\rho(N_{k}))\right)\leq\frac{1}{4}\mu_{n}^{{\mathbb{R}}}(B).$

###### Proof.

As in the proof of Lemma 17, we have that $\omega$ is uniformly distributed
and so for any $\varepsilon>0$ there exists $k_{\varepsilon}\in{\mathbb{N}}$
such that for all $k>k_{\varepsilon}$

$A_{N_{k}}(\omega,B):=\\#\left\\{1\leq j\leq N_{k}:\omega_{j}\in
B\right\\}\leq N_{k}(\mu_{n}^{{\mathbb{R}}}(B)+\varepsilon).$

Choose $\varepsilon<\mu_{n}^{{\mathbb{R}}}(B)$, and so

$A_{N_{k}}(\omega,B):=\\#\left\\{1\leq j\leq N_{k}:\omega_{j}\in
B\right\\}\leq 2N_{k}\mu_{n}^{{\mathbb{R}}}(B).$

for all $k>k_{\varepsilon}$. Then, by a standard covering argument we have
that

$\displaystyle\mu_{n}^{{\mathbb{R}}}\left(B\cap\bigcup_{1\leq j\leq
N_{k-1}}\Delta\left(\omega_{j},\rho(N_{k})\right)\right)$ $\displaystyle\leq
A_{N_{k-1}}(\omega,B)\mu_{n}^{{\mathbb{R}}}\left(\Delta\left(\omega_{j},\rho(N_{k})\right)\right)$
$\displaystyle=A_{N_{k-1}}(\omega,B)2^{n}v(N_{k})$ $\displaystyle\leq
N_{k-1}\mu_{n}^{{\mathbb{R}}}(B)2^{n+1}v(N_{k})$ $\displaystyle\leq
2^{-2}\mu_{n}^{{\mathbb{R}}}(B).$

where the last line follows from recalling that $N_{k-1}v(N_{k})\leq 2^{-n-3}$
∎

Combining the two lemmas we have that

$\displaystyle\mu_{n}^{{\mathbb{R}}}\left(B\cap\bigcup_{N_{k-1}\leq j\leq
N_{k}}\Delta\left(\omega_{j},\rho(N_{k})\right)\right)$
$\displaystyle\geq\mu_{n}^{{\mathbb{R}}}\left(B\cap\bigcup_{1\leq j\leq
N_{k}}\Delta\left(\omega_{j},\rho(N_{k})\right)\right)-\mu_{n}^{{\mathbb{R}}}\left(B\cap\bigcup_{1\leq
j\leq N_{k-1}}\Delta\left(\omega_{j},\rho(N_{k})\right)\right)$
$\displaystyle\geq\frac{1}{2}\mu_{n}^{{\mathbb{R}}}(B)-\frac{1}{4}\mu_{n}^{{\mathbb{R}}}(B)=\frac{1}{4}\mu_{n}^{{\mathbb{R}}}(B),$

as required. ∎

###  8.2. Proof of Theorem 15

By Proposition 8 and the definition of $({\mathcal{N}},v)$-d.s.s., we obtain a
ubiquitous system $(\\{R_{j}\\}_{j\in{\mathbb{N}}},\beta)$ with respect to
$({\mathcal{N}},v)$ when considering

  1. i.

$J={\mathbb{N}}$,

  2. ii.

$\beta:{\mathbb{N}}\to{\mathbb{N}}$, $\beta_{j}=j$ for all $j\in{\mathbb{N}}$,

  3. iii.

$N_{0}=1$ and $l_{j}=N_{j-1}$ and $u_{j}=N_{j}$ for $j\in{\mathbb{N}}$,

  4. iv.

$R_{j,i}=\omega_{j,i}$ and $R_{j}=\boldsymbol{\omega}_{j}\in[0,1]^{n}$

  5. v.

the function $v:{\mathbb{N}}\to{\mathbb{R}}_{+}$ given by $v(N)=D_{N}$ for all
$N\in{\mathbb{N}}$,

  6. vi.

Since our resonant sets are points we trivially have $\kappa_{i}=\kappa=0$.

The result follows from Theorem 5.

###  8.3. Proof of Theorem 16

Let $\tilde{v}:{\mathbb{N}}\to{\mathbb{R}}$ be the function given by

$\tilde{v}(N):=\sqrt{\frac{\log\log N}{2N}}\quad(N\in{\mathbb{N}}).$

Then, we have

$\tilde{v}\left(M^{3^{j}}\right)=\frac{j^{1/2}}{M^{3^{j}/2}}\sqrt{\frac{\log
3+\frac{\log\log M}{j}}{2}}\quad(j\in{\mathbb{N}}),$

so

$M^{3^{j-1}}v\left(M^{3^{j}}\right)\asymp_{M}M^{3^{j-1}}\frac{j^{1/2}}{M^{3^{j}/2}}=\frac{j^{1/2}}{(M^{1/6})^{3^{j}}}\to
0\;\text{ as }\;j\to\infty$

and

(8.6)
$\frac{\tilde{v}\left(M^{3^{j+1}}\right)}{\tilde{v}\left(M^{3^{j}}\right)}\asymp_{M}\frac{1}{M^{3^{j}}}\to
0\quad\text{ as }\quad j\to\infty.$

Observe that assuming (8.1), every large $j\in{\mathbb{N}}$ satisfies

$D_{M^{3^{j}}}^{*}<\frac{3}{2}\sqrt{\frac{\log\log M^{3^{j}}}{2M^{3^{j}}}}.$

For each $i\in\\{1,\ldots,n\\}$, define $\rho_{i}:{\mathbb{N}}\to{\mathbb{R}}$
by

$\rho_{i}(N)=\left(2^{n-1}3\tilde{v}(N)\right)^{\tau_{i}}\quad(N\in{\mathbb{N}}).$

Write $\omega=(\boldsymbol{\omega}_{j})_{j\geq 1}$,
$\omega_{j}=(\omega_{j,1},\ldots,\omega_{j,n})$ for all $j\in{\mathbb{N}}$,
and consider

  1. i.

$J={\mathbb{N}}$,

  2. ii.

$\beta_{j}=j$ for all $j\in{\mathbb{N}}$,

  3. iii.

For each $j\in{\mathbb{N}}$ put $R_{j,i}=\\{x_{j,i}\\}$ whenever $1\leq i\leq
n$ and $R_{j}=\\{\boldsymbol{\omega}_{j}\\}$,

  4. iv.

$u_{j}:=M^{3^{j}}$ and $l_{j}:=M^{3^{j-1}}$ for $j\in{\mathbb{N}}$,

  5. v.

$\kappa_{i}=0$ and $\delta_{i}=1$ for all $i\in\\{1,\ldots,n\\}$.

By Proposition 8, the pair $(\\{R_{\alpha}\\}_{\alpha\in J},\beta)$ is a
ubiquitous system with respect to $\rho$, $(l_{j})_{j\geq 1}$, and
$(u_{j})_{j\geq 1}$. Moreover, by (8.6), each $\rho_{i}$ is $c$-regular. The
result is a consequence of Theorem 5. For $n=1$ and $n=2$ we apply Lemma 2 for
$\psi_{i}(M^{3^{j}})\leq\rho_{i}(M^{3^{j}})$ to hold for large
$j\in{\mathbb{N}}$.

###  8.4. Proof of Theorem 17

Theorem 17 is shown as Theorem 16. In this case, however, we work with the
function $\tilde{v}:{\mathbb{N}}\to{\mathbb{R}}$ given by

$\tilde{v}(N)=\frac{(\log N)^{n}}{N}\quad(N\in{\mathbb{N}})$

and the sequences $l_{j}=M^{(j-1)^{2}}$, $u_{j}=M^{j^{2}}$ for all
$j\in{\mathbb{N}}$. In view of

$N_{j-1}\tilde{v}(N_{j})=M^{(j-1)^{2}}\frac{(j^{2}\log
M)^{n}}{M^{j^{2}}}=j^{2n}(\log M)^{n}M^{-(2j-1)}\to 0\quad\text{ as
}j\to\infty,$

we may define $v=C\tilde{v}$ for some positive constant $C=C(\omega)>0$ in
such a way that $\omega$ is a $((M^{j^{2}})_{j\geq 1},v)$-d.s.s. Finally, the
functions

(8.7) $\rho_{i}(N)=\left(\frac{(\log N)^{n}}{N}\right)^{\tau_{i}}\quad(1\leq
i\leq n)$

are non-increasing and tend to $0$. The theorem now follows from Proposition
8.

###  8.5. Proof of Theorems 19 and 20

####  8.5.1. Upper bound

For any $K\in{\mathbb{N}}$ consider the cover

$\bigcup_{j>K}\Delta(\omega_{j},\Psi(j))\supset W_{\omega}(\Psi)\,.$

Fix $1\leq k\leq n$. Each rectangle $\Delta(\omega_{j},\Psi(j))$ can be
covered by

$\asymp\prod_{i=1}^{n}\max\left\\{1,j^{\tau_{k}-\tau_{i}}\right\\}=j^{\sum_{i:\tau_{k}>\tau_{i}}(\tau_{k}-\tau_{i})}$

balls of radius $j^{-\tau_{k}}$. So

$\displaystyle\mathcal{H}^{s}\left(W_{\omega}(\Psi)\right)$
$\displaystyle\ll\sum_{j>K}j^{\sum_{i:\tau_{k}>\tau_{i}}(\tau_{k}-\tau_{i})}(j^{-\tau_{k}})^{s}$
$\displaystyle\leq\sum_{j>K}j^{-s\tau_{k}+\sum_{i:\tau_{k}>\tau_{i}}(\tau_{k}-\tau_{i})}\to
0$

as $K\to\infty$ for any

$s>\frac{1+\sum_{i:\tau_{k}>\tau_{i}}(\tau_{k}-\tau_{i})}{\tau_{k}}\,.$

Hence

$\operatorname{\dim_{H}}W_{\omega}(\Psi)\leq\frac{1+\sum_{i:\tau_{k}>\tau_{i}}(\tau_{k}-\tau_{i})}{\tau_{k}}$

Since the above argument remains true for each $1\leq k\leq n$ we have the
required upper bound. This gives the required upper bound for both Theorem 19
and 20.

####  8.5.2. Lower bound

As stated above, for any low discrepancy sequence we can associate the above
pair $({\mathcal{N}},v)$ to ensure that $\omega$ is a
$({\mathcal{N}},v)$-d.s.s and so by Proposition 8 $(\omega,\rho)$ is a local
ubiquitous system for functions (8.7) hence Theorem 6 is applicable in the
lower bound of Theorem 19. Note that strictly speaking the functions $\Psi$ in
Theorem 19 should be of the form

$\Psi(N)=\left(\left(\frac{(\log
N)^{n}}{N}\right)^{\tau_{1}},\dots,\left(\frac{(\log
N)^{n}}{N}\right)^{\tau_{n}}\right)\,.$

However, it is easily seen that for any $\theta>0$ and any
$0<\varepsilon<\theta$ we have

$\left(\frac{1}{N}\right)^{\theta}<\left(\frac{(\log
N)^{n}}{N}\right)^{\theta}<\left(\frac{1}{N}\right)^{\theta-\varepsilon}$

for $N$ sufficiently large, and so the Hausdorff dimension bound is the same.

From the conditions of Theorem 20 we can choose a sequence of natural numbers
over which $D_{N}(\omega)\ll N^{-1}$. Take a suitably sparse subsequence of
these integers to be ${\mathcal{N}}$ such that $N_{k-1}N_{k}^{-1}\to 0$ as
$k\to\infty$. Paired with the function $v(N)=cN^{-1}$, where $c$ is the
implied constant in the condition of Theorem 20, we have that $\omega$ is a
$({\mathcal{N}},v)$-d.s.s for such pair. Hence we have a weighted ubiquitous
system of rectangles for functions

$\rho_{i}(N)=\rho(N)^{a_{i}}=N^{-a_{i}}\quad(1\leq i\leq n)\,,$

for vector $\boldsymbol{a}=(a_{1},\dots,a_{n})\in{\mathbb{R}}_{+}$ such that

$\sum_{i=1}^{n}a_{i}=1\,.$

We now consider both lower bound results in tandem. It remains to show the
formula given in Theorem 6 produces the lower bound of Theorem 19 and 20. In
order to apply Theorem 6, for any
$\mathbf{a}=(a_{1},\dots,a_{n})\in{\mathbb{R}}_{+}$ such that
$\sum_{i=1}^{n}a_{i}=1$ and $\boldsymbol{\tau}=(\tau_{1},\dots,\tau_{n})$ as
in Theorem 19 and 20, let us define

$\textbf{t}=(t_{1},\dots,t_{n})=(\tau_{1}-a_{1},\dots,\tau_{n}-a_{n})\,.$

Consider the two cases:

  1. i.

Suppose that $\tau_{i}\geq\frac{1}{n}$ for all $1\leq i\leq n$. Then set

$a_{i}=\frac{1}{n}\quad(1\leq i\leq n)\,.$

Now, if $A=a_{i}$ for any $1\leq i\leq n$ we have that
$\mathcal{K}_{1}=\\{1,\dots,n\\}$ and so the lower bound formula is $n$ in
this case. If $A=\tau_{j}$ for some $1\leq j\leq n$ then

$\mathcal{K}_{1}=\emptyset\,,\quad\mathcal{K}_{2}=\\{i:\tau_{j}\geq\tau_{i}\\}\,,\quad\mathcal{K}_{3}=\mathcal{K}_{2}^{c}\,,$

and so

$\displaystyle\operatorname{\dim_{H}}W_{\omega}(\Psi)$
$\displaystyle\geq\min_{1\leq j\leq
n}\left\\{\\#\mathcal{K}_{2}+\frac{\sum_{i\in\mathcal{K}_{2}^{c}}a_{i}-\sum_{i\in\mathcal{K}_{2}}t_{k}}{\tau_{j}}\right\\}$
$\displaystyle\geq\min_{1\leq j\leq
n}\left\\{\\#\mathcal{K}_{2}+\frac{\sum_{i=1}^{n}a_{i}-\sum_{i\in\mathcal{K}_{2}}(a_{i}+t_{i})}{\tau_{j}}\right\\}$
$\displaystyle\geq\min_{1\leq j\leq
n}\left\\{\frac{\sum_{i=1}^{n}a_{i}+\sum_{i\in\mathcal{K}_{2}}(\tau_{j}-(a_{i}+t_{i})}{\tau_{j}}\right\\}$
$\displaystyle\geq\min_{1\leq j\leq
n}\left\\{\frac{1+\sum_{i:\tau_{j}\geq\tau_{i}}(\tau_{j}-\tau_{i})}{\tau_{j}}\right\\}.$

  2. ii.

Assume that there exists $\tau_{j}$ such that $\tau_{j}<\frac{1}{n}$. Then
without loss of generality suppose that

$\tau_{1}>\tau_{2}>\dots>\tau_{n}\,.$

We want to choose $1\leq u\leq n$ that solves

$u\times\widetilde{D}+\sum\limits_{u<i\leq n}\tau_{i}=1$

for some $\widetilde{D}>0$ with $\tau_{u}>\widetilde{D}$. Pick

$\widetilde{D}=\frac{1-\sum_{u\leq i\leq n}\tau_{i}}{u}$

and note that

$\tau_{1}>\tau_{2}>\dots>\tau_{u}>\widetilde{D}\geq\tau_{u+1}>\dots>\tau_{n}.$

Hence pick

$a_{i}=\begin{cases}\widetilde{D}\quad(1\leq i\leq u),\\\
\tau_{i}\quad(u+1\leq i\leq n).\end{cases}$

For $A=a_{i}$ with $1\leq i\leq u$ we have

$\mathcal{K}_{1}=\\{1,\dots,u\\},\quad\mathcal{K}_{2}=\\{u+1,\dots,n\\},\quad\mathcal{K}_{3}=\emptyset,$

and for $A=a_{i}=\tau_{i}$ with $u+1\leq i\leq n$

$\mathcal{K}_{1}=\\{1,\dots,i\\},\quad\mathcal{K}_{2}=\\{i+1,\dots,n\\},\quad\mathcal{K}_{3}=\emptyset.$

In both cases we have a lower bound formula of $n$.

For each $A=a_{j}+t_{j}=\tau_{j}$ with $1\leq j\leq u$ we have that

$\mathcal{K}_{1}=\emptyset,\quad\mathcal{K}_{2}=\\{j,\dots,n\\}=\\{i:\tau_{j}\geq\tau_{i}\\},\quad\mathcal{K}_{3}=\mathcal{K}_{2}^{c},$

Notice these are the same sets as in the conclusion of case $i)$, and so we
obtain the same lower bound, thus the proof of Theorem 19 and 20 is complete.

Acknowledgments: We thank Johannes Schleischitz and Dong Han Kim for useful
discussions.

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# Vehicular Blockage Modelling and Performance Analysis for mmWave V2V
Communications

Kai Dong, Marouan Mizmizi, Dario Tagliaferri and Umberto Spagnolini
Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB), Politecnico
di Milano, Milan, Italy.
Email: {kai.dong, marouan.mizmizi, dario.tagliaferri,
<EMAIL_ADDRESS>
###### Abstract

Vehicle-to-Everything (V2X) communications are revolutionizing the
connectivity of transportation systems supporting safe and efficient road
mobility. To meet the growing bandwidth eagerness of V2X services, millimeter-
wave (e.g., 5G new radio over spectrum 26.50 - 48.20 GHz) and sub-THz (e.g.,
120 GHz) frequencies are being investigated for the large available spectrum.
Communication at these frequencies requires beam-type connectivity as a
solution for the severe path loss attenuation. However, beams can be blocked,
with negative consequences for communication reliability. Blockage prediction
is necessary and challenging when the blocker is dynamic in high mobility
scenarios such as Vehicle-to-Vehicle (V2V). This paper presents an analytical
model to derive the unconditional probability of blockage in a highway multi-
lane scenario. The proposed model accounts for the traffic density, the 3D
dimensions of the vehicles, and the position of the antennas. Moreover, by
setting the communication parameters and a target quality of service, it is
possible to predict the signal-to-noise ratio distribution and the service
probability, which can be used for resource scheduling. Exhaustive numerical
results confirm the validity of the proposed model.

###### Index Terms:

V2V communication, dynamic blockage, millimeter-waves, sidelink

##  I Introduction

Recent advances in Vehicle-to-Everything (V2X) communications accelerated the
evolution of intelligent transportation systems towards a safe and efficient
traffic management [1]. Currently deployed V2X technologies operates at sub-6
GHz frequencies [2] and can meet the requirements of basic V2X services due to
the limited available bandwidth. Future 6G V2X is foreseen to support a wide
range of services [3] exploring high frequencies in the Millimeter Wave
(mmWave) or even sub-THz bandwidths [4]. However, the wireless propagation at
these frequencies is subject to severe path-loss attenuation. Collimated
communication and beamforming technology is a viable solution to increase the
communication range, but mmWave/sub-THz beams are subject to frequent
blockage, which in vehicular scenarios such as Vehicle-to-Vehicle (V2V) links,
represents the killing issue. According to the Third Generation Partnership
Project (3GPP) [5], there are four main propagation conditions: (i) Line-of-
Sight (LoS); (ii) blockage due to a building (NLoSb); (iii) blockage due to
foliage (NLoSf) and (iv) blockage due to vehicles (NLoSv). In this setting,
blockage modeling and prediction are essential for the proper design of robust
V2X systems to enable the critical requirements of ultra-high reliability and
ultra-low latency.

Blockage impact on V2X communication has been receiving increasing attention
in the recent literature. In [6], V2V channel measurements in urban and
highway roads considering different frequency bands in the presence of vehicle
blockage were experimentally analyzed. At $30$ GHz, the blockage from a
vehicle causes a power loss ranging from $10$ to $20$ dB [7], depending on the
transmitting vehicle (TxV) and receiving vehicle (RxV) position, and it
increases by $5.5-17$ dB when multiple vehicles are simultaneously blocking
the LoS. Moreover, based on an extensive measurement campaign [8], the 3GPP
suggests modeling the vehicle blockage loss as a lognormal random variable
added to the path loss, while in [9] the a-priori probabilities of LoS and
NLoSv are derived without considering the traffic density. The authors in [10]
extended the 3GPP model including the NLoSb condition and three traffic
density levels, i.e., low, medium, and high. A further extension of the
previous works is in [11], considering the empirical blockage probabilities in
highway and urban scenarios. However, the number of blocking vehicles and the
variability of their shapes, particularly of the height, is not considered.
The authors of [12] derived the end-to-end outage probability assuming the
presence of a single blocking vehicle with a height modelled as a Gaussian
distribution. However, the case of multiple vehicles simultaneously blocking
the LoS has never been considered, including the contribution of the NLoSv to
the communication link.

This paper addresses this gap in the literature by presenting an analytical
model of SNR distribution and the service probability in highway V2V scenarios
considering the impact of (i) multiple blocking vehicles, (ii) random
vehicles’ height, and (iii) traffic density. The path-loss and blockage models
are based on the 3GPP recommendations, and the vehicular traffic is obtained
by extending the work in [13, 14], where the traffic distribution is
represented by a Poisson Point Process (PPP). This analytical tool can be
useful in designing solutions for reliable and robust V2X networks and for
resource scheduling to mitigate the blockage effects. The presented models are
validated through extensive numerical simulations with vehicular traffic
generator under various traffic conditions.

The remainder of this paper is organized as follows: Section II presents the
system model and SNR derivation. The proposed vehicle blockage is derived in
Section III, while Section IV shows the numerical validation. Finally, closing
remarks are discussed in Section V.

##  II System Model

Figure 1: A linear highway scenario for sidelink V2V communications.

The considered scenario is a multi-lane highway road segment, depicted in Fig.
1, where vehicles are randomly distributed with density $\rho$ [veh/m].
Vehicles are equipped with a mmWave/sub-THz V2V communication interface
mounted either on the rooftop or on the frontal/rear bumper [9], and are
assumed to have a box-shaped occupation of size $w_{v}\times l_{v}\times
h_{v}$, as shown in Fig. 1.

For an arbitrary V2V link between a transmitting vehicle (TxV) and a receiving
vehicle (RxV), the propagation path-loss (in dB) is [9]

$\begin{split}PL(k)&=\underbrace{32.4+20\log_{10}(d_{tr})+20\log_{10}(f_{c})}_{\mu_{\text{LoS}}}+A(k)+\chi\end{split}$
(1)

where $d_{tr}$ is the distance between the TxV and RxV, $f_{c}$ is the carrier
frequency (in GHz), $\chi\sim\mathcal{N}(0,\sigma_{sh}^{2})$ represents the
lognormal distributed shadowing component, and
$A(k)\sim\mathcal{N}(\mu(k),\sigma^{2}(k))$ accounts for an additional
attenuation due to blockage from $k$ vehicles. Assuming independence between
the shadowing component and the blockage component, we can write the path-loss
as

$PL(k)\sim\mathcal{N}\left(\underbrace{\mu_{\text{LoS}}+\mu(k)}_{\mu_{PL}(k)},\underbrace{\sigma_{sh}^{2}+\sigma^{2}(k)}_{\sigma_{PL}^{2}(k)}\right)$
(2)

where $\mu_{\text{LoS}}$ is the deterministic term in (1).

The signal-to-noise ratio (SNR) $\gamma(k)$ (in dB) is

$\gamma(k)=P_{t}+G_{t}+G_{r}-PL(k)-P_{n}$ (3)

where $P_{t}$ is the transmitted power, $G_{t}$ and $G_{r}$ are the TxV and
RxV array gains, respectively, $P_{n}$ is the noise power. The SNR
distribution for distance $d_{tr}$ and for $k$ vehicle blockers at the LoS V2V
link is

$f_{\gamma^{(k)}}(\gamma|d_{tr})=\frac{1}{\sqrt{2\pi\sigma^{2}_{PL}(k)}}\exp\left\\{-\frac{(\gamma-\mu_{\gamma}(k))^{2}}{2\,\sigma_{PL}^{2}(k)}\right\\}$
(4)

with mean value $\mu_{\gamma}(k)=P_{t}+G_{t}+G_{r}-\mu_{PL}(k)-P_{n}$ and
variance $\sigma_{PL}^{2}(k)$, both dependent on the propagation condition
between TxV and RxV.

##  III Vehicle Blockage Modelling and Performance Evaluation

Figure 2: First Fresnel Ellipsoid blockage characterization considering the
height of the vehicles

From the electromagnetic perspective, the LoS propagation is affected when one
(or more) blocking vehicle obstructs the first Fresnel ellipsoid. Let’s assume
the height of each vehicle as a Gaussian random variable
$h_{v}\sim\mathcal{N}(\mu_{v},\sigma_{v}^{2})$ [15] and a single blocker
$\mathcal{B}$ at distance $d_{tb}$ from TxV and $d_{br}$ from RxV (along the
LoS line), as depicted in Figure 2. We can use the knife edge theory on link
propagation [16] to express the effective height of the first Fresnel
ellipsoid at the blocker position as

$\tilde{h}=(h_{r}-h_{t})\frac{d_{tb}}{d_{tr}}+h_{t}-0.6\,\tilde{r}$ (5)

where $h_{t}$ and $h_{r}$ are the heights of the TxV and RxV, respectively and
$\tilde{r}$ is the radius of the first Fresnel ellipsoid at relative distance
$d_{tb}$ and $d_{br}$

$\tilde{r}=\sqrt{\lambda_{c}\frac{d_{tb}d_{br}}{d_{tb}+d_{br}}}.$ (6)

with $\lambda_{c}$ being the carrier wavelength. For the propagation at mmWave
frequencies ($30-300$ GHz), the term $\tilde{r}$ might be meaningful in (5).
For a V2V system operating in 5G NR FR2 bands ($28$ GHz) and a maximum TxV-RxV
distance $d_{tr}=200$ m, $\tilde{r}\approx 1$ m and therefore must be taken
into account. Differently, for sub-THz systems ($>100$ GHz) and shorter
distances ($<50$ m), $\tilde{r}$ can be reasonably neglected. Hence, the
Fresnel ellipsoid height is, in general,

$\tilde{h}=h_{r}\frac{d_{tb}}{d_{tr}}+h_{t}\frac{d_{br}}{d_{tr}}-0.6\,\tilde{r}\sim\mathcal{N}\left(\tilde{\mu},\tilde{\sigma}^{2}\right)$
(7)

with mean $\tilde{\mu}=\mu_{v}-0.6\,\tilde{r}$ and variance
$\tilde{\sigma}^{2}=\sigma_{v}^{2}$, assuming the independence between the TxV
and RxV heights $h_{t}$ and $h_{r}$. Thus, a blockage occurs when the random
blocker height $h_{b}\sim\mathcal{N}(\mu_{b},\sigma_{b}^{2})$ is larger than
$\tilde{h}$, i.e., $h_{eff}=h_{b}-\tilde{h}>0$. The probability of having a
blockage given the presence of a vehicle $\mathcal{B}$ between the TxV and RxV
can be computed as:

$\mathbb{P}\left(\text{NLoSv}|d_{tr},\mathcal{B}\right)=Q\left(\frac{h_{eff}-\mu_{eff}}{\sigma_{eff}}\right)$
(8)

where $\mu_{eff}=\mu_{b}-\tilde{\mu}$,
$\sigma_{eff}^{2}=\sigma^{2}_{b}+\tilde{\sigma}^{2}$, and $Q(x)$ is the
Gaussian Q-function.

The probability of having $k\geq 0$ blockers depends on the distribution of
vehicles in the scenario and on the distance $d_{tr}$ between TxV and RxV. To
ease the analytical derivation, we first determine the probability of having
$k$ blockers within a given distance $d_{tr}$, and subsequently, we will
remove this constraint obtaining the unconstrained SNR distribution.

In a $M$-lane highway V2V scenario, the probability of having a vehicle on any
arbitrary lane is $1/M$. Therefore, the probability of having both TxV and RxV
on the same lane is $M(1/M^{2})=1/M$, while the probability of having them on
different lanes is $1-(1/M)$. In the two cases, the maximum number of $k$
blockers is different.

(a) Same lane

(b) Different lanes

Figure 3: Proposed vehicle blockage modelling approaches: (a) both TxV and RxV
locate at the same lane; (b) TxV and RxV in different lanes.

###  III-A TxV and RxV on the Same Lane

When the TxV and RxV are on the same lane, as shown in Figure 3a, the maximum
number of blockers, for example in a traffic jam, is $N_{s}=\lfloor
d_{eff}/d_{a}\rfloor$, where the size of the vehicle occupancy is

$d_{a}=l_{v}+d_{s},$ (9)

that is the effective length of the vehicle incorporating a safety distance
$d_{s}$, while $d_{eff}=d_{tr}-l_{v}$ is the effective distance between TxV
and RxV. Dividing the lane into $N_{s}$ vehicle occupancy slots of size
$d_{a}$, the probability of having a simultaneous blockage by $k$ vehicles is
thus given by the probability of having $k$ occupied slots out of $N_{s}$,
that is given by the Bernoulli probability

$\mathbb{P}_{SL}\left(\text{NLoSv}^{(k)}|d_{tr}\right)=\begin{pmatrix}N_{s}\\\
k\end{pmatrix}\mathcal{P}_{a}^{k}(1-\mathcal{P}_{a})^{N_{s}-k}$ (10)

where $\mathcal{P}_{a}$ is the probability that a single slot is occupied by a
blocker. This latter term is

$\begin{split}\mathcal{P}_{a}=\mathbb{P}\left(\text{NLoSv}|d_{a},\mathcal{B}\right)\,\mathbb{P}\left(\mathcal{B}\right)=Q\left(\frac{h_{eff}-\mu_{eff}}{\sigma_{eff}}\right)\Gamma\,e^{-\Gamma},\end{split}$
(11)

assuming that vehicles on a lane are distributed according to a linear Point
Poisson Process (PPP) as in [13][14], with $\Gamma=\rho\,d_{a}$.

###  III-B TxV and RxV on Different Lanes

Assuming the TxV and RxV are on two distinct lanes, the blocking vehicles can
occupy the same lane as the TxV and/or RxV or different lanes for $M>2$, as
shown in Figure 3b. Similar to the previous setup, we divide the area between
TxV and RxV into $\widetilde{M}$ possible occupancy slots (where
$\widetilde{M}$ denotes the amount of spanned lanes by the V2V link) that, if
occupied by a vehicle, could block the LoS. The size of the occupancy slots
determines the blockage probability, and we can identify two types, i.e.,
blockage from the same lane of the TxV or RxV and different lanes. The sizes
can be derived geometrically, that for the first type is

$d_{b}=\frac{w_{v}\sqrt{d_{tr}^{2}-\Delta y^{2}}}{2\Delta y}$ (12)

and for the second type is

$d_{c}=\frac{w_{v}\sqrt{d_{tr}^{2}-\Delta y^{2}}}{\Delta y}+l_{v}$ (13)

where $\Delta y=n\,W$ is the lateral displacement among the lanes. The
probability of having a blockage in occupancy slot type $b$ and $c$, namely
$\mathcal{P}_{b}$ and $\mathcal{P}_{c}$, respectively, can be computed as in
(11), with $\Gamma=\rho d_{b}$ and $\Gamma=\rho d_{c}$. Since $d_{b}$ and
$d_{c}$ depend on $n\,W$ for $n>1$ (for $n=0$, the TxV and RxV are on the same
lane), we need to consider all possible displacements separately, i.e.,
$n=1,...,M-1$. For $n=1$, the maximum number of blocking slots is
$\widetilde{M}=2$ with an occupancy slot size computed as in (12). The
probability of having $k$ blockers out of $2$ can be computed using the
Bernoulli probability (10). For $n>1$, the maximum number of blocking slots is
$\widetilde{M}=n+1$. In this case, we have two slots’ types with sizes
computed as in (12) and (13). The probability of having $k$ blockers out of
$n+1$ is obtained by Poisson Binomial

$\mathbb{P}(K=k)=\sum_{\mathcal{A}\in\mathcal{Q}_{k}}\prod_{i\in\mathcal{A}}\mathcal{P}_{i}\prod_{j\in\mathcal{A}^{c}}(1-\mathcal{P}_{j})$
(14)

where $\mathcal{Q}_{k}$ is the set of all $k$ slots that can be selected from
$\\{1,\dots,n+1\\}$ and $\mathcal{A}^{c}$ is the complement of $\mathcal{A}$,
i.e., $\mathcal{A}^{c}=\\{1,\dots,n+1\\}\not\in\mathcal{A}$. For example, when
$n=2$ and $k=2$, then $\mathcal{Q}_{2}=\\{\\{1,2\\},\\{2,3\\},\\{1,3\\}\\}$.
The number of combination in $\mathcal{Q}_{k}$ is given by the binomial
coefficient

$\begin{pmatrix}n+1\\\ k\end{pmatrix}=\frac{(n+1)!}{(n-k+1)!k!}$ (15)

To combine all the cases for different $n$, we need to compute the probability
of having TxV and RxV with lateral displacement $n\,W$, that is

$\mathbb{P}\left(\Delta y=n\,W\right)=2\frac{M-n}{M^{2}}$ (16)

assuming that the probability of having a vehicle on an arbitrary lane is
$1/M$.
Remark: by considering all possible combination of $n=\\{1,\dots,M-1\\}$, the
aggregated can be written as

$\frac{2}{M^{2}}+\sum_{n=1}^{M-1}n=\frac{M-1}{M}$ (17)

knowing that $\sum_{i=1}^{m}i=m(m+1)/2$. By considering the same lane setup,
i.e., $n=0$, the sum of the probabilities of all events equals 1.

The probability of having $k$ simultaneous vehicle blockers when TxV and RxV
are on different lanes is derived in (18), while the overall blockage
probability constrained on having TxV and RxV at a distance $d_{tr}$ is in
(19).

$\mathbb{P}_{DL}\left(\text{NLoSv}^{(k)}|d_{tr}\right)=\frac{2(M-1)}{M^{2}}\begin{pmatrix}n+1\\\
k\end{pmatrix}\mathcal{P}_{b}^{k}(1-\mathcal{P}_{b})^{n+1-k}+\sum_{n=2}^{M-1}\frac{2(M-n)}{M^{2}}\sum_{\mathcal{A}\in\mathcal{Q}_{k}}\prod_{i\in\mathcal{A}}\mathcal{P}_{i}\prod_{j\in\mathcal{A}^{c}}(1-\mathcal{P}_{j})$
(18)

$\mathbb{P}\left(\text{NLoSv}^{(k)}|d_{tr}\right)=\mathbb{P}_{DL}\left(\text{NLoSv}^{(k)}|d_{tr}\right)+\frac{1}{M}\mathbb{P}_{SL}\left(\text{NLoSv}^{(k)}|d_{tr}\right)$
(19)

To obtain the unconstrained probability of blockage, the probability density
function $f_{D}(d)$ of the distance
$d_{tr}=\sqrt{(x_{r}-x_{t})^{2}+(y_{r}-y_{t})^{2}}$ is required. $f_{D}(d)$
has been widely studied in geometric probability [17] and can be derived as in
[18], assuming that $x_{r}\sim x_{t}\sim\mathcal{U}\\{0,D\\}$ with $D$ the
maximum length of the considered highway scenario, and that $y_{t}$ and
$y_{r}$ are discrete uniform random variables with $p(y)=1/M$.

###  III-C Signal-to-Noise Ratio Distribution

The SNR probability density function $f_{\gamma^{(k)}}(\gamma|d_{tr})$ in (4)
is Gaussian distributed with $\mu_{\gamma}(k)$ and $\sigma_{\gamma}^{2}(k)$
that depend on the number of blocking vehicles $k$. The unconstrained SNR
distribution results in a Gaussian mixture, representing the different
propagation conditions:

$\begin{split}f_{\gamma}(\gamma|d_{tr})=&\,\mathbb{P}(\text{LoS}|d_{tr})\,f_{\gamma^{(0)}}(\gamma|d_{tr})\,+\\\
&+\sum_{k=1}^{B}\mathbb{P}(\text{NLoSv}^{(k)}|d_{tr})\,f_{\gamma^{(k)}}(\gamma|d_{tr})\end{split}$
(20)

where
$\mathbb{P}(\text{LoS}|d_{tr})=1-\sum_{k}\mathbb{P}(\text{NLoSv}^{(k)}|d_{tr})$
is the LoS probability, and $B=\mathrm{max}(N_{s},M)$ is the maximum possible
number of blockers.

##  IV Numerical Simulations

This section presents the numerical validation of the proposed analytical
framework. In particular, we derive the SNR distribution in (20) considering
different scenario setups, and we analyze the communication performances in
terms of service probability:

$\begin{split}P_{S}=\mathbb{P}(\gamma\geq\gamma_{th})&=\mathbb{P}(\text{LoS}|d_{tr})\,F_{\gamma^{(0)}}(\gamma_{th})+\\\
&+\sum_{k=1}^{B}\mathbb{P}(\text{NLoSv}^{(k)}|d_{tr})\,F_{\gamma^{(k)}}(\gamma_{th}),\end{split}$
(21)

where $F_{\gamma^{(k)}}(\gamma_{th})$ is the normal cumulative distribution
function. Unless otherwise specified, the simulation parameters in Table I are
used. These parameters are selected based on the 3GPP recommendation in [9],
and similarly for the antenna configuration, as depicted in Figure 4.

TABLE I: Parameter Setups Parameter | Symbol | Value
---|---|---
Transmit power | $P_{t}$ |  $0$ dBm
Transmit antenna gain | $G_{t}$ |  $10$ dB
Receive antenna gain | $G_{r}$ |  $10$ dB
Noise Power | $P_{n}$ |  $-85$ dBm
Vehicle length, width | $[l_{v},w_{v}]$ |  $[5,1.8]$ m
Vehicle height mean and std. dev. | $\mu_{h,v},\sigma_{h,v}$ |  $[1.5,0.08]$ m
Min. inter-vehicle safety distance | $d_{s}$ |  $2.5$ m
Scenario length | $D$ |  $200$ m
Number of lanes | $M$ | $3$
Figure 4: Rooftop vs bumper antenna configuration according to 3GPP [9]

(a)

(b)

Figure 5: Blockage probability and average number of vehicle blockers varying
the TxV and RxV inter-distance $d_{tr}$: (a) Blockage probability; (b) average
number of blockers.

(a)

(b)

Figure 6: SNR distribution fitted with Monte Carlo simulations, conditioned to
distance $d_{tr}$ for both TxV and RxV in the same lane: (a) different
distance $d_{tr}$ and density $\rho$, and (b) different antenna configurations
(i.e., rooftop and bumper antenna setups).

Exhaustive Monte Carlo simulations are carried out to validate the proposed
analytical model. Figure 5 (a) depicts the blockage probability varying the
distance $d_{tr}$ between TxV and RxV and considering two traffic densities,
namely $\rho=10$, and $50$ veh/Km. This result shows that for high distances,
the blockage probability attains around $50$% and $90$% in low and high
traffic densities, respectively. This can also be observed in Figure 5 (b),
where, for the same scenario, the average number of blockers is reported. The
proposed model is used to predict the communication performances by deriving
the SNR distribution in (4) and the service probability in (21). Figure 6
shows the predicted SNR distribution under various traffic densities and
distances. In particular, for the considered setting, we observe that the
average SNR in Figure 6 (a) is below $0$ dB for high traffic density due to
the higher blockage probability. Figure 6 (b) depicts the predicted SNR
distribution when considering the antenna mounted on the rooftop and the
bumper in the same scenario. As expected, the antennas on the bumper
experience a loss of approximately $4.2$ dB in SNR. This result suggests
considering rooftop antenna design for better communication performances as it
shows more robustness against blockage.

The predicted SNR distribution can be used to derive the service probability
using (21) for a given QoS, i.e., SNR threshold $\gamma_{th}$. Figure 7
depicts the service probability varying both traffic density rho and distance
$d_{tr}$ for an SNR threshold $\gamma_{th}=0$ dB. Even at low density, we have
a poor service probability; therefore, direct V2V communication can support
data rate demanding applications only for a limited range. This trend is more
severe when the antennas are mounted on the bumper, as shown in Figure 8 (a),
and when increasing the QoS, as shown in Figure 6 (b).

The analytical tools presented in this paper can be valuable in designing
solutions that increase the reliability and robustness of future V2X networks,
for example, scheduling resources for relaying from other vehicles or roadside
units to mitigate the blockage effect.

Figure 7: Service probability constrained by a inter-vehicle distance $d_{tr}$
under different traffic densities based on the derived SNR in (20) constrained
by $\gamma_{th}=0$ dB.

(a)

(b)

Figure 8: Service probability evaluation ($d_{tr}=50$ m) varying with (a)
density $\rho$ for cases of TxV and RxV in the same lane and neighbour lanes
($\gamma_{th}=0$ dB); (b) SNR threshold $\gamma_{\text{th}}$ constrained by
different service categories.

##  V Conclusion

In 5G/6G beam-based V2V communications operating at mmWave/sub-THz
frequencies, the dynamic LOS link blockage due to obstructing vehicles is the
dominant source of SNR loss, hindering the development of advanced safety-
critical services in the context of autonomous driving. The vehicle-induced
blockage depends on the number of potential vehicles blocking the LOS link as
well as on their sizes. This paper proposes a novel analytical blockage
modelling method, deriving a closed-form SNR distribution that takes into
account the traffic density, the random 3D occupancy region of each vehicle, a
minimum inter-vehicle safety distance, and separately accounts for the case in
which TxV and RxV are located on the same lane or on different lanes.
Moreover, as indicated by 3GPP, we account for both rooftop and bumper antenna
placements. Numerical simulations validate the proposed analytical model,
showing a good match between the analytical and the simulated SNR in all
conditions. The presented analytical model is proved to be a tool for
predicting the performance of V2V networks under blockage from dynamic objects
such as surrounding vehicles. A significant service probability loss is
observed when TxV and RxV are located on the same lane, and when the antenna
arrays are mounted on the bumper, suggesting the usage of a rooftop-mounted
antenna equipment. The results also show that V2V communication at mmW/sub-THz
can only support short-range applications, while other solutions, e.g.,
relaying from other vehicles, are needed to increase the range without
outages.

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