On sums of $k$-th powers with almost equal primes

Wei Zhang

Abstract For “almost all” sufficiently large $N,$ satisfying necessary
congruence conditions and $k\geq 2$, we show that there is an asymptotic
formula for the number of solutions of the equation

$\displaystyle\begin{split}&N=p_{1}^{k}+p_{2}^{k}+\cdots+p_{s}^{k},\\\
&\left|p_{i}-(N/s)^{1/k}\right|\leq(N/s)^{\theta/k},\ (1\leq i\leq
s)\end{split}$

with

$\displaystyle s\geq\frac{k(k+1)}{2}+1\ \ \textup{and}\ \ \theta\geq{\bf
2/3}+\varepsilon.$

This enlarges the effective range of $s$ for which can be obtained by the
method of Mätomaki and Xuancheng Shao [12]. [Discorrelation between primes in
short intervals and polynomial phase, Int. Math. Res. Not. IMRN 2021, no. 16,
12330-12355.]

The idea is to avoid using the exponential sums (1.2) and Vinogradov mean
value theorems in Lemma 2.4 simultaneously. And the main new ingredient is
from Kumchev and Liu [9] (see Lemma 2.2).

Keywords Waring-Goldbach problem; exponential sums over primes in short
intervals; circle method.

2000 Mathematics Subject Classification 11P32, 11P05, 11L20, 11P55

##  1\. Introduction

For $k\in\mathbb{Z}^{+}$ and a prime $p,$ let $\tau=\tau(k,p)$ be the largest
integer such that $p^{\tau}|k.$ Define

$\eta(k,p)=\tau+2$

when $p=2$ and $\tau>0,$ and $\eta(k,p)=\tau+1$ otherwise. Then define

$R=R(k)=\prod_{(p-1)|k}p^{\eta}.$

For sufficiently large $N,$ $N\in\mathcal{N}_{s}=\\{N\in\mathbb{N}:N\equiv
s(\mathrm{mod}\ R)\ \textup{and}\ 9\nmid N,\ \textup{for}\ k=3\ \textup{and}\
s=7\\}$,

$\displaystyle\begin{split}&N=p_{1}^{k}+p_{k}^{k}+\cdots+p_{s}^{k},\\\
&\left|p_{i}-(N/s)^{1/k}\right|\leq(N/s)^{\theta/k},\ (1\leq i\leq
s)\end{split}$ (1.1)

was investigated by many experts. For example, see [8, 9, 10, 12, 13, 15]. In
[8], this was first studied for $s\geq 7$ and $k=2$. Recently, this was
improved by many authors. For example, see [9, 12, 13, 15].

In [9], Kumchev and Liu proved that for sufficiently large $N,$
$N\in\mathcal{N}_{s}=\\{N\in\mathbb{N}:N\equiv s(\mathrm{mod}\ R)\\},$ there
are solutions of the equation (1.1) with $s\geq k^{2}+k+1$ and $\theta\geq
31/40+\varepsilon,$ which gives an asymptotic lower bound of the correct order
for the number of the representations. They also proved that “almost all”
sufficiently large $N,$ $N\in\mathcal{N}_{s}=\\{N\in\mathbb{N}:N\equiv
s(\mathrm{mod}\ R)\\}$, (1.1) is solvable with $s\geq\frac{k(k+1)}{2}+{\bf 1}$
and $\theta\geq 31/40+\varepsilon.$ Next we will introduce the definition of
“almost all”. For primes $p_{i}$, define

$E(N,s,x^{k-1}y)=|\mathcal{E}(N,s,x^{k-1}y)|,$

where

$\displaystyle\mathcal{E}(N,s,x^{k-1}y)$
$\displaystyle=\left\\{n:n\in(N,N+x^{k-1}y]\cap\mathcal{N}_{s},\
\varrho(n)-\mathfrak{S}(n)\mathfrak{J}(n)\geq y^{s-1}x^{1-k}(\log
x)^{-1}\right\\},$

where $\varrho(n)$ is the number of representations of $n$ in the form of
(1.1) and $\mathfrak{S}(n),\mathfrak{J}(n)$ are defined by ($\ref{asy}).$ If

$\lim_{N\longrightarrow\infty}\frac{E(N,s,x^{k-1}y)}{x^{k-1}y}=0\ \
\left(\textup{i.e.}\ E(N,s,x^{k-1}y)=o\left(x^{k-1}y\right)\right),$

then we say that for “almost all” sufficiently large $N,$
$N\in\mathcal{N}_{s},$ there is an asymptotic formula for the number of
solutions of the equation (1.1) with certain $s$ and $\theta.$

Recently, these results were improved by Matomäki-Shao [12] and Salmensuu [13]
respectively. Matomäki-Shao [12] use the exponential sums. In order to use the
exponential sums

$\displaystyle\sum_{x<n\leq x+x^{2/3+\varepsilon}}\Lambda(n)e(\alpha n^{k}),$
(1.2)

they need more variables to apply a better type Vinogradov mean value theorem
related to [3] instead of Lemma 2.4. By using the Harman sieve, Salmensuu got
some even better results. Unfortunately, both their results do not contain the
situation of “almost all” for $s\leq k^{2}+k$. However, the method of
Matomäki-Shao [12] implies that “almost all” sufficiently large $N,$
$N\in\mathcal{N}_{s}=\\{N\in\mathbb{N}:N\equiv s(\mathrm{mod}\ R)\\},$ there
is an asymptotic formula for the number of solutions of the equation (1.1)
with $s\geq(k^{2}+k)/2+2$ and $\theta\geq 2/3+\varepsilon.$ This improves the
previous results for the aspect of $\theta$ but with worse $s$ (as cited in
[12]). The aim of this paper is to reduce the $\frac{k(k+1)}{2}+{\bf 2}$ to
$\frac{k(k+1)}{2}+{\bf 1}.$ Then the range of $s$ is the same as previous
results in [15, 9]. The idea is to avoid using the exponential sums (1.2) and
the Vinogradov mean value theorems in Lemma 2.4 simultaneously.

######  Theorem 1.1.

For “almost all” sufficiently large $N,$
$N\in\mathcal{N}_{s}=\\{N\in\mathbb{N}:N\equiv s(\mathrm{mod}\ R)\\},$ there
is an asymptotic formula for the number of solutions of the equation

$\displaystyle\begin{split}&N=p_{1}^{k}+p_{2}^{k}+\cdots+p_{s}^{k},\\\
&\left|p_{i}-(N/s)^{1/k}\right|\leq(N/s)^{\theta/k},\ (1\leq i\leq
s)\end{split}$

with

$s\geq\frac{k(k+1)}{2}+1\ \ \textup{and}\ \ \ \theta\geq 2/3+\varepsilon.$

In fact, such an idea can also be used to enlarge the effective range of $s$
of [12]. Though there are some other results in [13] with much better
$\theta.$ However, such type results cannot be used to give something related
to asymptotic formulas for the number of solutions of the equation (1.1). By
similar arguments, one can get the following corollary.

######  Corollary 1.2.

For all sufficiently large $N,$ $N\in\mathcal{N}_{s}=\\{N\in\mathbb{N}:N\equiv
s(\mathrm{mod}\ R)\\},$ there is an asymptotic formula for the number of
solutions of the equation

$\displaystyle\begin{split}&N=p_{1}^{k}+p_{2}^{k}+\cdots+p_{s}^{k},\\\
&\left|p_{i}-(N/s)^{1/k}\right|\leq(N/s)^{\theta/k},\ (1\leq i\leq
s)\end{split}$

with

$s\geq k(k+1)+1\ \ \textup{and}\ \ \ \theta\geq 2/3+\varepsilon.$

##  2\. Preliminaries

For $v\in\mathbb{Z}^{+},$ let $w_{k}(q)$ be defined as

$\displaystyle w_{k}(p^{ku+v})=\begin{cases}kp^{-u-1/2}&\textup{for}\ \ u\geq
0\ \textup{and}\ \ v=1,\\\ p^{-u-1}&\textup{for}\ \ u\geq 0\ \textup{and}\ \
2\leq v\leq k.\end{cases}$ (2.1)

Then we will introduce two lemmas. These lemmas are from [16] (in Chinese) and
[9] respectively.

######  Lemma 2.1.

Let $\log D\ll\log x$ and $1\leq\max\\{x^{1/2},M\\}\leq y\leq x.$ Then there
exist a constant $c_{0}>0$ uniformly for $\gamma\in\mathbb{R}$ such that

$\sum_{q\leq M}\sum_{\mbox{\tiny$\begin{array}[]{c}1\leq a\leq q\\\ (a,q)=1\\\
\end{array}$}}\int_{|\alpha-a/q|\leq 1}\frac{w_{k}(q)^{2}|\sum_{x<p\leq
x+y}(\log p)e(p^{k}(\alpha+\gamma))|^{2}}{1+D|\alpha-a/q|}d\alpha\ll
y^{2}D^{-1}\log^{c_{0}}x.$

###### Proof.

One can refer to the case $k=3$ and $k=4$ in [17] and [18], respectively. And
this is Lemma 2.1 in [16] (in Chinese). So we only sketch the proof. For
$\alpha=a/q+\gamma_{1},$ for the properties of geometric series, we have

$\sum_{1\leq a\leq q}\left|\sum_{x<p\leq x+y}(\log
p)e(p^{k}(\alpha+\gamma))\right|^{2}\leq q\sum_{x<p_{1},\ p_{2}\leq x+y\atop
p_{1}^{k}\equiv p_{2}^{k}(\textup{mod}q)}(\log p_{1})(\log p_{2}).$

On the other hand, we have the estimate

$q\sum_{x<p_{1},\ p_{2}\leq x+y,\ p_{1}^{k}\equiv p_{2}^{k}(\textup{mod}\
q)}1\ll y^{2}d(q)^{c_{k,1}}.$

As (see Lemma 2.1 in [19])

$\sum_{q\leq M}w_{k}(q)^{2}d(q)^{c_{k,2}}\ll(\log M)^{c_{k,3}}.$

Then the desired conclusion follows. For similar proofs, one can refer to [2,
1, 17, 18, 19]. ∎

Next Lemma is from [9].

######  Lemma 2.2.

Suppose that $y=x^{\theta},$ $0<\rho\leq t_{k}^{-1},$
$\frac{1}{2-t_{k}\rho}\leq\theta\leq 1$ and $A:=(x,x+y]$, where

$t_{k}=\left\\{\begin{array}[]{ll}2,&\textup{if}\ k=2,\\\
k^{2}-k+1,&\textup{if}\ k\geq 3.\end{array}\right.$ (2.2)

Then either

$\displaystyle\sum_{n\in A}e(\alpha n^{k})\ll y^{1-\rho+\varepsilon},$ (2.3)

or there exist integers $a$ and $q$ such that

$\displaystyle 1\leq q\leq y^{k\rho},\ \ (a,q)=1,\ \ |q\alpha-a|\leq
x^{1-k}y^{k\rho-1},$ (2.4)

and

$\displaystyle\sum_{n\in A}e(\alpha
n^{k})\ll\frac{{\color[rgb]{0,0,1}w_{k}(q)}y}{1+yx^{k-1}|\alpha-a/q|}+y^{1-\rho+\varepsilon}.$
(2.5)

###### Proof.

Similar as the proof in [9, 4, 3], but estimate

$I(\beta)=\int_{x-y}^{x+y}e(\beta\gamma^{k})d\gamma$

in another way (as the way in [7]), where $\beta=\alpha-a/q.$ By Lemma 6.2 in
[14], we have

$I(\beta)\ll w_{k}(q)y(1+|\beta|yx^{k-1})^{-1}$

instead of the estimates in [9] and [3], i.e.

$I(\beta)\ll y(q+q|\beta|yx^{k-1})^{-1/k}.$

Combining the analysis in [9, 4], we finally complete the proof. ∎

######  Lemma 2.3 (see [12], see also in [11] for the case of $k=2$).

Let $y=x^{\theta}$ for some fixed $\theta\geq 2/3+\varepsilon.$ Let
$\alpha\in\mathbb{R}$ and let $k$ be a positive integer. Suppose that

$\left|\sum_{x<n\leq x+y}\Lambda(n)e(\alpha n^{k})\right|\geq\frac{y}{(\log
x)^{A}}$

for some $A\geq 2.$ Then there exists a positive integer $q\leq(\log
x)^{O_{k}(A)}$ such that

$|q\alpha-a|x^{k-1}y\leq(\log x)^{O_{k}(A)}.$

######  Lemma 2.4 (see [9]).

For $y\geq x^{1/2+\varepsilon},$ $s\geq k^{2}+k$ and $k\geq 2,$ we have

$\int_{[0,1]}\left|\sum_{x<n\leq x+y}(\log p)e(\alpha
p)\right|^{s}\mathrm{d}\alpha\ll y^{s-1}x^{1-k+\varepsilon}.$

Next we will prove another lemma to achieve our goals. The proof involves some
ideas in [6, 17, 19].

######  Lemma 2.5.

Suppose that $y=x^{\theta},$ $0<\rho\leq t_{k}^{-1},$
$\frac{1}{2-t_{k}\rho}\leq\theta\leq 1.$ For $\mathcal{I},$ a subinterval of
$(x,x+y],$ let $h(\alpha)=h_{\mathcal{I}}(\alpha)=\sum_{n\in\mathcal{I}}(\log
n)e(\alpha n^{k}).$ For $1\leq q\leq y^{k\rho},$ $(a,q)=1,$
$\mathcal{M}_{q,a}=\\{\alpha:|q\alpha-a|\leq x^{1-k}y^{k\rho-1}\\},$ let
$\mathcal{M}$ be the union of the intervals of $\mathcal{M}(q,a).$ Then for
any measurable set $\mathfrak{W}\subseteq[0,1],$ we have

$\int_{\mathfrak{W}}|h(\alpha)|^{s}d\alpha\ll y^{2}(\log
x)^{2}\mathcal{J}_{0}^{1/2}\left(\int_{\mathfrak{W}}|h(\alpha)|^{2s-6}\right)^{1/2}+y^{2-\rho+\varepsilon}\mathcal{J}(\mathfrak{W})$

where

$\mathcal{J}_{0}=\sup_{\beta\in[0,1]}\int_{\mathcal{M}}\frac{w_{k}^{2}(q)|h(\alpha\pm\beta)|^{2}}{\left(1+x^{k-1}y|\alpha-a/q|\right)^{2}}d\alpha$

and

$\mathcal{J}(\mathfrak{W})=\int_{\mathfrak{W}}|h(\alpha)|^{{\bf s-2}}d\alpha.$

###### Proof.

Write

$\mathcal{M}_{\alpha}=\mathcal{M}_{\alpha}(q,a)-\beta=\\{\beta:|q\alpha-q\beta-a|\leq
x^{1-k}y^{k\rho-1}\\}$

and

$\mathcal{M}_{\alpha}=\bigcup_{q\leq y^{k\rho}}\bigcup_{1\leq a\leq q,\
(a,q)=1}\mathcal{M}_{\alpha}(q,a).$

Following the argument in [17], one can get

$\displaystyle\begin{split}\left|\int_{\mathfrak{W}}|h(\alpha)|^{s}d\alpha\right|^{2}&=\left|\sum_{n\in\mathcal{I}}(\log
n)\int_{\mathfrak{W}}e(\alpha
n^{k})|h(\alpha)|^{s-2}h(-\alpha)d\alpha\right|^{2}\\\ &\ll\left|\log
x\sum_{x<n\leq x+y}\left|\int_{\mathfrak{W}}e(\alpha
n^{k})|h(\alpha)|^{s-2}h(-\alpha)d\alpha\right|\right|^{2}.\end{split}$

By Cauchy’s inequality, we have

$\displaystyle\begin{split}\left|\int_{\mathfrak{W}}|h(\alpha)|^{s}d\alpha\right|^{2}\ll
y(\log x)^{2}\sum_{x<n\leq x+y}\left|\int_{\mathfrak{W}}e(\alpha
n^{k})|h(\alpha)|^{s-2}h(-\alpha)d\alpha\right|^{2}.\end{split}$ (2.6)

Expand the square, we can get

$\displaystyle\sum_{x<n\leq x+y}$
$\displaystyle\left|\int_{\mathfrak{W}}e(\alpha
n^{k})|h(\alpha)|^{s-2}h(-\alpha)d\alpha\right|^{2}$
$\displaystyle=\int_{\mathfrak{W}}\int_{\mathfrak{W}}\sum_{x<n\leq
x+y}e((\alpha-\beta)n^{k})|h(\alpha)|^{s-2}h(-\alpha)|h(\beta)|^{s-2}h(\beta)d\alpha
d\beta.$

Write

$f(\gamma)=\sum_{x<n\leq x+y}e(\gamma n^{k}).$

Note that

$|h(-\alpha)h(\beta)|\leq|h(\alpha)|^{2}+|h(\beta)|^{2}.$

Then

$\displaystyle\sum_{x<n\leq x+y}$
$\displaystyle\left|\int_{\mathfrak{W}}e(\alpha
n^{k})|h(\alpha)|^{s-2}h(-\alpha)d\alpha\right|^{2}$
$\displaystyle\ll\int_{\mathfrak{W}}\int_{\mathfrak{W}}\left|f(\alpha-\beta)\right|\left(\left|h(\alpha)\right|^{s-2}|h(\beta)|^{s}\right)d\alpha
d\beta.$

Thus

$\displaystyle\int_{\mathfrak{W}}\int_{\mathfrak{W}}\left|f(\alpha-\beta)\right|\left|h(\alpha)\right|^{s-2}|h(\beta)|^{s}d\alpha
d\beta$
$\displaystyle\ll\int_{\mathfrak{W}}\int_{\mathfrak{W}\cap\mathcal{M}_{\beta}}|h(\beta)|^{s}|h(\alpha)|^{s-2}\frac{w_{k}(q)y}{1+x^{k-1}y|\alpha-\beta-a/q|}d\alpha
d\beta$
$\displaystyle+y^{1-\rho+\varepsilon}\int_{\mathfrak{W}}|h(\alpha)|^{s-2}d\alpha\int_{\mathfrak{W}}|h(\alpha)|^{s}d\alpha.$

Write

$T_{\beta}=\int_{\mathfrak{W}\cap\mathcal{M}_{\alpha}}|h(\beta)|^{s-2}\frac{w_{k}(q)y}{1+x^{k-1}y|\alpha-\beta-a/q|}d\beta.$

By Cauchy’s inequality, we have

$\displaystyle T_{\beta}$ $\displaystyle\ll
y\left(\int_{\mathfrak{W}}|h(\beta)|^{2s-6}d\beta\right)^{1/2}\left(\int_{\mathcal{M}}\frac{w_{k}^{2}(q)|h(\alpha-\beta)|^{2}}{\left(1+x^{k-1}y|\beta-a/q|\right)^{2}}d\beta\right)^{1/2}$
$\displaystyle\ll
y\left(\int_{\mathfrak{W}}|h(\beta)|^{2s-6}d\beta\right)^{1/2}\mathcal{J}_{0}^{1/2}.$

Hence

$\displaystyle\begin{split}\sum_{x<n\leq x+y}\left|\int_{\mathfrak{W}}e(\alpha
n^{k})|h(\alpha)|^{s-2}h(-\alpha)d\alpha\right|^{2}&\ll
y\int_{\mathfrak{W}}|h(\alpha)|^{s}d\alpha\left(\int_{\mathfrak{W}}|h(\alpha)|^{2s-6}d\alpha\right)^{1/2}\mathcal{J}_{0}^{1/2}\\\
&+y^{1-\rho+\varepsilon}\int_{\mathfrak{W}}|h(\alpha)|^{s-2}d\alpha\int_{\mathfrak{W}}|h(\alpha)|^{s}d\alpha.\end{split}$
(2.7)

Now the desired result can be deduced from (2.6) and (2.7).

∎

##  3\. Proof of Theorem 1.1

Let $x=(N/s)^{1/k}$ and $y=x^{\theta},$ where $N$ is a sufficiently large
natural number. Let $n$ satisfy $|n-N|\leq yx^{k-1}.$ Let

$f(\alpha)=\sum_{x-y<n\leq x+y}(\log p)e(\alpha p^{k}).$

denote the summation for primes $p.$ Then

$\varrho(n)=\int_{[0,1]}f(\alpha)^{s}e(-n\alpha)d\alpha.$

Let

$P=(\log x)^{A},\ \ \ \ Q=xy^{k-1}P^{-1}.$

Write

$I(q,a)=\\{\alpha\in[0,1):|q\alpha-a|\leq Q^{-1}\\}.$

Then define the major arcs $\mathfrak{M}$ and minor arcs $\mathfrak{m}$ as

$\mathfrak{M}=\mathfrak{M}(P,Q)=\bigcup_{q\leq
P}\bigcup_{\mbox{\tiny$\begin{array}[]{c}1\leq a\leq q\\\ (a,q)=1\\\
\end{array}$}}I(q,a),$
$\mathfrak{m}=\mathfrak{m}(P,Q)=[0,1)\setminus\mathfrak{M}.$

Suppose that $\mathfrak{B}$ is a measurable subset of $[0,1)$. Let

$\varrho(n,\mathfrak{B})=\int_{\mathfrak{B}}f(\alpha)^{s}e(-n\alpha)d\alpha.$

Then we can write

$\varrho(n)=\varrho(n,\mathfrak{M})+\varrho(n,\mathfrak{m}).$

The analysis of $\varrho(n,\mathfrak{M})$: We follow the analysis in [12] and
[15]. The width of our major arc is chosen so that if
$\alpha\in\mathfrak{M}(q,a),$ then $f(\alpha)$ can be estimated by counting
primes in short intervals in residue classes modulo $q.$ Since $\theta>7/12,$
we can use Huxley’s result on primes in short intervals to get

$f(a/q+\beta)=\phi(q)^{-1}S(q,a)v(\beta)+O(y/(\log)^{10})$

for

$|\beta|\leq P/(yx^{k-1}),$

where

$S(q,a)=\sum_{1\leq b\leq q\atop(b,q)=1}e\left(\frac{ab^{k}}{q}\right)$

and

$v(\beta)=k^{-1}\sum_{(x-y)^{k}\leq m\leq(x+y)^{k}}m^{-1+1/k}e(\beta m).$

Thus the standard theory of the major arc contributions in the Waring-Goldbach
problem can be applied to yield the estimate

$\displaystyle\varrho(n,\mathfrak{M})=\mathfrak{S}(n)\mathfrak{J}(n)+O\left(\frac{y^{s-1}}{x^{k-1}(\log
x)^{10}}\right),$ (3.1)

where $\mathfrak{S}(n)$ is the singular series

$\mathfrak{S}(n)=\sum_{q=1}^{\infty}\phi(q)^{-s}\sum_{1\leq a\leq
q\atop(a,q)=1}S(q,a)^{s}e(-an/q),$

and $\mathfrak{J}(n)$ is the singular integral

$\mathfrak{J}(n)=\int_{[0,1]}v(\beta)^{s}e(-\beta n)d\beta.$

On the other hand,

$\mathfrak{S}(n)\asymp 1,\ \ \ \mathfrak{J}(n)\asymp y^{s-1}x^{1-k}.$

Then we have

$\varrho(n,\mathfrak{M})\gg y^{s-1}x^{1-k}.$ (3.2)

$E(N,s,x^{k-1}y)$ does not exceed the number of integers $n$ for which

$\varrho(n,\mathfrak{m})\ll y^{s-1}x^{1-k}(\log x)^{-2}.$

Thus it follows from Bessel’s inequality that

$E(N,s,x^{k-1}y)\ll(y^{s-1}x^{1-k})^{-2}(\log
x)^{4}\int_{\mathfrak{m}}|f(\alpha)|^{2s}d\alpha,$

and theorem follows from the following proposition. Then I would state and
prove the proposition, and that will close the paper.

######  Proposition 3.1.

Let $A_{0}\geq 1$ be fixed and

$K(t)=\int_{\mathfrak{m}}|f(\alpha)|^{t}d\alpha.$

For $t\geq k^{2}+k+2$ and $y\geq x^{2/3+\varepsilon},$ we have

$K(t)\ll y^{t-1}x^{1-k}(\log x)^{-A_{0}},$

provided that $A$ is sufficiently large in terms of $A_{0}.$

###### Proof.

Fix $A_{0}.$ Choose $B=A_{0}+c_{0}+10.$ Choose $A$ sufficiently large in terms
of $B.$ Then Lemma 2.3 implies that for $y\geq x^{2/3+\varepsilon},$

$\sup_{\alpha\in\mathfrak{m}}|f(\alpha)|\leq y(\log x)^{-B}.$ (3.3)

Choose $h(\alpha)=f(\alpha)$ and $\mathfrak{W}=\mathfrak{m}$ in Lemma 2.5.
Then we have

$K(t)\ll y^{2}(\log
x)^{2}\mathcal{J}_{0}^{1/2}K(2t-6)^{1/2}+y^{2-\rho+\varepsilon}K(t-2).$

(i) For the first term, we have

$K(t)\ll y^{2}(\log x)^{2}\mathcal{J}_{0}^{1/2}K(2t-6)^{1/2}\ll y^{2}(\log
x)^{2}\mathcal{J}_{0}^{1/2}K(t)^{1/2}\max_{\alpha\in[0,1]}f(\alpha)^{(t-6)/2}.$

Hence

$K(t)^{2}\ll y^{4}(\log
x)^{4}\mathcal{J}_{0}K(t)\max_{\alpha\in[0,1]}f(\alpha)^{(t-6)}.$

Choosing $D=yx^{k-1}$ in Lemma 2.1, this gives

$K(t)\ll y^{4}(\log
x)^{4+c_{0}}(y^{2}(yx^{k-1})^{-1})\max_{\alpha\in[0,1]}f(\alpha)^{(t-6)}.$

Then by (3.3), we have

$K(t)\ll y^{t-1}x^{1-k}(\log x)^{-A_{0}}.$

(ii) For the second term, by the Vinogradov mean value theorem (see Lemma 2.4)

$\int_{[0,1)}f^{s}(\alpha)\ \mathrm{d}\alpha\ll y^{s-1}x^{1-k+\varepsilon},\ \
s\geq k^{2}+k,$

we have

$K(t)\ll y^{t-3}x^{1-k+\varepsilon}y^{2-\rho+\varepsilon}\ll
y^{t-1-\rho+\varepsilon}x^{1-k}.$

Choosing $\varepsilon>0$ sufficiently small to ensure that $\rho=4\varepsilon$
is admissible in Lemma 2.2. Then we can complete the proof. ∎

By the standard argument, we can get

$E(N,s,x^{k-1}y)\ll(y^{s-1}x^{1-k})^{-2}K(2s).$

Then Theorem 1.1 can be deduced by the standard argument, (3.2) and
Proposition 3.1.

$\mathbf{Acknowledgements}$ I am deeply grateful to the referee(s) for
carefully reading the manuscript and giving corrections, useful suggestions.

## References

  * [1] J. Brüdern, A sieve approach to the Waring-Goldbach problem I. Ann. Sci. Ecole Norm. Sup. Paris 28(1995) 461-476.
  * [2] J. Brüdern, A sieve approach to the Waring-Goldbach problem II: on the seven cubes theorem. Acta Arith. 72(1995) 211-227.
  * [3] D. Daemen, The asymptotic formula for localized solutions in Waring’s problem and approximations to Weyl sums. Bull. Lond. Math. Soc. 42(2010) 75-82.
  * [4] B. R. Huang, Exponential sums over primes in short intervals and applications in Waring-Goldbach problem. Mathematika 62(2016) 208-523.
  * [5] K. Kawada and T. D. Wooley, On the Waring-Goldbach problem for fourth and fifth powers. Proc. London Math. Soc. 83(2001) 1-50.
  * [6] K. Kawada, L. L. Zhao, The Waring-Goldbach problem for cubes with an almost prime. Proc. Lond. Math. Soc. 119(2019) 867-898.
  * [7] A.V.Kumchev, On Weyl sums over primes in short intervals, in Number Theory: Arithmetic in Shangri-La, Series on Number Theory and its Applications, Vol. 8 (World Scientific, Singapore, 2013), pp. 116-131.
  * [8] A.V. Kumchev, T.Y. Li, Sums of almost equal squares of primes. J. Number Theory 132(2012) 608-636.
  * [9] A. V. Kumchev, H. F. Liu, On sums of powers of almost equal primes. J. Number Theory 176(2017) 344-364.
  * [10] J. Y. Liu, G. S. Lü, T. Zhan, Exponential sums over primes in short intervals. Sci. China Ser. A 49 (2006) 611-619.
  * [11] G. S. Lü, H.X. Lao, On exponential sums over primes in short intervals. Monatsh. Math. 151(2007) 153-164.
  * [12] K. Matömaki, X. C. Shao, Discorrelation between primes in short intervals and polynomial phase. arXiv:1902.04708.
  * [13] J. Salmensuu, On the Waring-Goldbach problem with almost equal summands. arXiv:1903.01824.
  * [14] R.C. Vaughan, The Hardy-Littlewood method, section edition, Cambridge University Press, Cambridge 1997.
  * [15] B. Wei, T. D. Wooley, On sums of powers of almost equal primes. Proc. London. Math. Soc. 111(2015) 1130-1162.
  * [16] Y. J. Yao, Waring-Goldbach problems with almost equal variables and related problems, PhD Thesis, Shandong University (2015) (in Chinese).
  * [17] Y. J. Yao, Sums of nine almost equal prime cubes. Front. Math. China 9(2014) 1131-1140.
  * [18] Y. J. Yao, M. Zhang, Waring-Goldbach problem for fourth powers with almost equal variables. Lith. Math. J. 56(2016) 572-581.
  * [19] L. L. Zhao, On the Waring-Goldbach problem for fourth and sixth powers. Proc. London. Math. Soc. 108(2014) 1593-1622.


# Data-to-text Generation for Severely Under-Resourced Languages
with GPT-3.5: A Bit of Help Needed from Google Translate

Michela Lorandi  Anya Belz
ADAPT Research Centre
Dublin City University, Ireland
<EMAIL_ADDRESS>
###### Abstract

LLMs like GPT are great at tasks involving English which dominates in their
training data. In this paper, we look at how they cope with tasks involving
languages that are severely under-represented in their training data, in the
context of data-to-text generation for Irish, Maltese, Welsh and Breton.
During the prompt-engineering phase we tested a range of prompt types and
formats on GPT-3.5 and 4 with a small sample of example input/output pairs. We
then fully evaluated the two most promising prompts in two scenarios: (i)
direct generation into the under-resourced language, and (ii) generation into
English followed by translation into the under-resourced language. We find
that few-shot prompting works better for direct generation into under-
resourced languages, but that the difference disappears when pivoting via
English. The few-shot + translation system variants were submitted to the
WebNLG 2023 shared task where they outperformed competitor systems by
substantial margins in all languages on all metrics. We conclude that good
performance on under-resourced languages can be achieved out-of-the box with
state-of-the-art LLMs. However, our best results (for Welsh) remain well below
the lowest ranked English system at WebNLG’20.

##  1 Introduction

Pretrained large language models (LLMs) are everywhere, achieving state-of-
the-art-results across many application areas. LLMs display strong out-of-the-
box abilities at a wide range of different tasks far beyond language-based
tasks. However, these abilities have been explored mostly for English which
dominates text online, hence the data on which LLMs are trained. In the work
reported here, we explore to what extent state-of-the-art LLMs can help with
tasks involving severely under-resourced languages which by definition would
have been very scarce in their training data.

More specifically, we address the challenging task of data-to-text generation
for four severely under-resourced languages: Irish, Maltese, Welsh and Breton.
Developing any new system is difficult in a severely under-resourced scenario,
more so where the task has language-specific input/output combinations as in
data-to-text and text-to-text.

The paper is structured as follows. Section 2 describes data and task, Section
3 presents the general approach, prompt engineering, and the specific systems
we fully evaluated. Experimental set-up and results are outlined in Section 4,
and Section 5 provides discussion and conclusions.

All code and results are available on Github: https://github.com/DCU-NLG/DCU-
NLG-PBN.

##  2 Data and Task

The WebNLG’23 Aquilina et al. (2023) data consists of 1,778 dev items for each
language, 1,399 test items for Breton, and 1,665 for Welsh, Irish and Maltese,
that were manually translated by professional translators from the English
originals. Additionally 13,211 training items are provided where texts were
automatically translated from English.

As in all WebNLG shared tasks, WebNLG 2023 systems must map from the data (RDF
triples) to a suitable output text, as in the example from the WebNLG’23
website111https://synalp.gitlabpages.inria.fr/webnlg-challenge/docs in Figure
1. This is a data-to-text generation task which is well established in the
natural language generation (NLG) field.

Figure 1: WebNLG example of input triples and output text in English.

##  3 General Approach and Specific Systems

The general approach we take is to use a state-of-the-art LLM in its out-of-
the-box state, to either (i) generate texts directly from the data in the
target language, or (ii) generate texts in English and then translate them
into the target language. More specifically, the LLM we used was the
InstructGPT model identified as text-davinci-003, and the MT system was Google
Translate which covers all WebNLG’23 languages except Breton.

In all cases, we fed a prompt representing the generation task to text-
davinci-003, and collected the system completion as the output which we then
cleaned up by simple post-processing procedures.

Figure 2: The two prompt types used in the final systems. Given Input
(triples) highlighted in yellow, model output in blue. The few-shot in-context
prompt also incorporates examples (highlighted green).33footnotemark: 3

###  3.1 Prompt testing and selection

We conducted three prompt testing phases, in each phase using a different
random sample of 20 data/text pairs, stratified for triple size, with outputs
in all target languages, from the dev set. These small subsets of the dev set
represent all the data we used during system development.

Phase 1: In the first prompt testing phase, we tested seven prompt variants on
text-davinci-003 on all four target languages and a subset of four prompts on
GPT-4 on Irish and Breton. The prompts were of three types: zero-shot minimal
instruction, few-shot in-context learning, and chain-of-thought Wei et al.
(2022).

On GPT-3.5, we tested each prompt in two scenarios: (i) generating the texts
directly into the target languages, (ii) generating all texts into English and
then translating them into each target language. In all prompts, the input
triples were surrounded by triple quotes as per best-practice recommendations;
we tested removing these quotes for the zero-shot prompt. On GPT-4, we tested
zero-shot and few-shot with and without translation, and chain-of-thought
(CoT) without translation.

The clearest results immediately were that (i) generation into English plus
translation worked better than direct generation into the target languages,
and (ii) the CoT prompts performed far worse than the others, a difference
that was exacerbated when using GPT-4. Zero-shot looked better for Irish, but
few-shot had the edge for the other three languages.

BLEU, METEOR and ChrF++ for zero-shot looked overall slightly better on GPT-4,
but TER was substantially worse and moreover the emerging patterns were less
clear. Leaving out the triple quotes did not appear to work better (although
this was tested on only one language).

Phase 2: As we were testing on a very small sample, the risk was poor
representativeness of the data as a whole. Therefore, in the second prompt
testing phase, we used a fresh sample of 20 pairs to further test the most
promising zero-shot and few-shot prompts in both +/- translation scenarios. We
dropped the chain-of-thought prompts and GPT-4, as the Phase 1 results had
been less promising.

Results confirmed better performance for few-shot, this time in all scenarios
with all evaluation metrics (although for a small number of results, scores
were virtually the same). Again having quotes worked better than leaving them
out.

At this point we tested a postprocessing method on most of the previous
experiments. This revealed clear improvements in almost all cases. The method
replaces HTML with ASCII characters (e.g. &quot is replaced with ").
Furthermore, we remove underscores, and quotes that wrap the entire text.

Phase 3: In the third prompt testing phase, we tested just the zero-shot and
few-shot prompts into English with translation, on Irish and Welsh. At this
stage the aim was to further investigate the difference between the two so far
most promising types of prompts, on another fresh data sample, on these two
languages because results were most similar for different prompt variants for
them. However, Phase 3 results for zero-shot and few-shot + MT for Irish and
Welsh were again overall tied.

###  3.2 Prompt types used in final systems

Prompt testing had revealed zero-shot and few-shot prompts with triple quotes
generating into English plus MT as clearly the most promising. Figure 3 shows
examples of these two prompt variants in terms of the prompt template, few-
shot examples (where applicable), and system completion. For more example
prompts, see appendix, Section A. Zero-shot minimal instruction prompts are
simple requests to perform a task without details or examples, in our case
consisting of a brief sentence describing the task and the input to the task.
Few-shot in-context learning prompts are composed of a brief description of
the task and a list of examples providing both inputs and target outputs,
ending in just an input, for which the system response is taken to be the
output.

The final prompt formats follow best practice for prompt engineering provided
by OpenAI;444https://help.openai.com/en/articles/6654000-best-practices-for-
prompt-engineering-with-openai-api input triples are recast as subject
predicate object.

###  3.3 System variants evaluated

In the full evaluations on the WebNLG’23 test set reported below, we tested
our selected zero-shot and few-shot prompts in both +/- MT scenarios, giving
us four system variants as follows:

  1. 1.

Zero-shot generation into each language;

  2. 2.

Few-shot generation into each language;

  3. 3.

Zero-shot generation into English + translation into each language;

  4. 4.

Few-shot generation into English + translation into each language.

##  4 Experimental Set-up and Results

We executed our experiments using the paid-for OpenAI API to access text-
davinci-003,555https://platform.openai.com/docs/models/gpt-3-5 and the free
Google Translate API,666https://cloud.google.com/translate in early June 2023.

In all experiments, we set text-davinci-003 parameters to temperature=0, top
p=1 (default) to consider all possible tokens, frequency penalty=0 and
presence penalty=0 (default value) to slightly reduce repetitive tokens, best
of=1 (default) to get only 1 completion for each prompt, and maximum length to
500. All generated texts are post-processed as described above.

Language | Method |  BLEU $\uparrow$ |  ChrF++ $\uparrow$ |  TER $\downarrow$
---|---|---|---|---
Irish | Zero-Shot Irish | 12.9931 | 0.4124 | 0.9298
Few-Shot Irish | 15.3477 | 0.4303 | 0.8451
Zero-Shot English + Google Translate | 20.5176 | 0.5146 | 0.7122
Few-Shot English + Google Translate | 20.4001 | 0.51 | 0.6894
WebNLG 2023 Baseline | 11.63 | 0.36 | 0.74
Maltese | Zero-Shot Maltese | 13.0311 | 0.445 | 0.8496
Few-Shot Maltese | 15.4315 | 0.4536 | 0.7605
Zero-Shot English + Google Translate | 20.3528 | 0.5263 | 0.67
Few-Shot English + Google Translate | 21.2656 | 0.5249 | 0.6465
WebNLG 2023 Baseline | 15.60 | 0.42 | 0.67
Welsh | Zero-Shot Welsh | 15.8695 | 0.4619 | 0.822
Few-Shot Welsh | 18.9512 | 0.4742 | 0.7192
Zero-Shot English + Google Translate | 24.7126 | 0.5496 | 0.6659
Few-Shot English + Google Translate | 25.115 | 0.5484 | 0.6435
WebNLG 2023 Baseline | 10.70 | 0.36 | 0.77
Breton | Zero-Shot Breton | 15.2498 | 0.4217 | 0.82
Few-Shot Breton | 17.207 | 0.4394 | 0.739
WebNLG 2023 Baseline | 9.92 | 0.33 | 0.76
Table 1: Results for our system variants (Section 3), and the WebNLG’23
baseline results provided by the organisers. Highest score in each column for
each language in boldface.

Table 1 presents BLEU Papineni et al. (2002), ChrF++ Popović (2017) and TER
Snover et al. (2006) for our four systems, and the baseline reported by the
WebNLG organisers Aquilina et al. (2023): the winning system for English from
WebNLG’20 combined with the Zero MT system.

Comparing zero-shot results with corresponding few-shot results, it is clear
that the addition of examples to the prompt helps the model perform the task
better in all languages: BLEU and TER are always higher for the few-shot
variant than the zero-shot variant, while ChrF++ is always higher for few-shot
where translation is not used, and virtually identical where translation is
used.

When generating the texts in English plus translation, we don’t see the same
strong benefit achieved by the addition of few-shot examples noted above. Even
where there is a benefit, it is small. In fact, we observe similar performance
between zero-shot + MT and few-shot + MT throughout, while both clearly
outperform versions without translation; this is the case in all three
languages where we have translation (Irish, Maltese, and Welsh).

It might be speculated that providing few-shot examples partially compensates
for the comparatively smaller amount of knowledge that the LLM has about the
under-resourced languages. This would explain why the benefit disappears when
pivoting via English, where examples can add little to the knowledge the LLM
already has.

##  5 Discussion and Conclusion

We submitted the three few-shot + MT systems to the WebNLG Challenge
2023,777https://synalp.gitlabpages.inria.fr/webnlg-
challenge/challenge_2023/docs in the Welsh, Irish and Maltese tracks. Based on
the metric evaluation results (test set) shared by WebNLG organisers, these
systems outperform all other submitted systems by sizeable margins; moreover,
few-shot Breton without translation (∗) also outperforms the one submitted
system substantially:

| Few-Shot + GTransl | Nearest system
---|---|---
| BLEU | chrF++ | TER | BLEU | chrF++ | TER
Irish | 20.4 | 0.51 | 0.69 | 16.66 | 0.44 | 0.75
Maltese | 21.27 | 0.52 | 0.65 | 16.49 | 0.47 | 0.7
Welsh | 25.12 | 0.55 | 0.64 | 20.97 | 0.49 | 0.67
Breton∗ | 17.21 | 0.44 | 0.74 | 13.11 | 0.33 | 0.76

At the time of writing, we do not know what kinds of systems produced the
runner-up results, except that the Irish runner-up is the handcrafted rule-
based FORGe system extended for Irish Mille et al. (2023). There are
interesting differences between FORGe and our system. FORGe took a
considerable amount of person time, but uses no paid-for resources. To create
and test our system took next to no time, and US$91.82 in API costs. For
FORGe, the system and how it was created is known. For our system, we don’t
have access to the model itself, and only incomplete knowledge about its exact
technical details at the particular time we access it; on another day, results
may differ. This makes experiments non-repeatable, and generalising to future
use of the same approach difficult. Moreover, it’s not a system that can be
deployed in this form, because of the changing nature of what’s at the other
end of the LLM and MT APIs.

This raises important general questions about (i) how scientific research is
to be conducted with LLMs served as-a-service that deliver best performance,
but whose technical details aren’t known and moreover change over time; and
(ii) how such research results can be utilised in developing systems for real-
world deployment.

In this paper we have explored different prompt types and formats for data-to-
text generation for under-resourced languages, with out-of-the-box LLMs with
and without machine translation. We found that few-shot + MT achieves new
state-of-the-art metric performance at the WebNLG’23 RDF-to-text generation
task for the severely under-resourced languages Irish, Maltese, Welsh and
Breton. However, given the caveats above, results presented here must be
treated as a rough indication of what is achievable with an LLM+MT approach.

## Acknowledgements

The cost of accessing the GPT API was covered by financial support from the
DCU-NLG Research Group at DCU. Michela Lorandi’s work was conducted with the
financial support of the Science Foundation Ireland Centre for Research
Training in Digitally-Enhanced Reality (d-real) under Grant No. 18/CRT/6224.
Both authors benefit from being members of the ADAPT SFI Research Centre at
Dublin City University, funded by the Science Foundation Ireland under Grant
Agreement No. 13/RC/2106_P2. For the purpose of Open Access, the author has
applied a CC BY public copyright licence to any Author Accepted Manuscript
version arising from this submission.

## References

  * Aquilina et al. (2023) Enrico Aquilina, Anya Belz, Claudia Borg, Liam Cripwell, Claire Gardent, Albert Gatt, John Judge, Michela Lorandi, Anna Nikiforoskaya, William Soto-Martinez, and Craig Thomson. 2023.  The 2023 webnlg shared task on low resource languages overview and evaluation results (webnlg 2023).  In _Proceedings of the Workshop on Multimodal, Multilingual Natural Language Generation and Multilingual WebNLG Challenge_ , Prague, Czech Republic.
  * Mille et al. (2023) Simon Mille, Elaine U’i Dhonnchadha, Stamatia Dasiopoulou, Lauren Cassidy, Brian Davis, and Anya Belz. 2023.  DCU/TCD-FORGe at WebNLG’23: Irish rules!  In _Proceedings of the Workshop on Multimodal, Multilingual Natural Language Generation and Multilingual WebNLG Challenge_ , Prague, Czech Republic.
  * Papineni et al. (2002) Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. 2002.  Bleu: a method for automatic evaluation of machine translation.  In _Proceedings of the 40th annual meeting of the Association for Computational Linguistics_ , pages 311–318.
  * Popović (2017) Maja Popović. 2017.  chrF++: words helping character n-grams.  In _Proceedings of the Second Conference on Machine Translation_ , pages 612–618, Copenhagen, Denmark. Association for Computational Linguistics.
  * Snover et al. (2006) Matthew Snover, Bonnie Dorr, Rich Schwartz, Linnea Micciulla, and John Makhoul. 2006\.  A study of translation edit rate with targeted human annotation.  In _Proceedings of the 7th Conference of the Association for Machine Translation in the Americas: Technical Papers_ , pages 223–231, Cambridge, Massachusetts, USA. Association for Machine Translation in the Americas.
  * Wei et al. (2022) Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Ed Chi, Quoc Le, and Denny Zhou. 2022.  Chain of thought prompting elicits reasoning in large language models.  _arXiv preprint arXiv:2201.11903_.

##  Appendix A Prompts

We provide all the prompts used in our experiments, specifying the template
and one example for each prompt. In Table 2 we provide the zero-shot minimal
instruction prompt, while in Table 3 we provide the few-shot in-context prompt
with the same set of examples we used in our experiments.

Zero-Shot minimal instruction
---
Template: |  Write the following triples as fluent English | Irish | Welsh | Maltese | Breton text. Triples: """ {set of triples in the format subject predicate object and each triple in a new line} """ Text: [MODEL]
Example Prompt: |  Write the following triples as fluent English text. Triples: """ AC_Hotel_Bella_Sky_Copenhagen owner Bella_Center AC_Hotel_Bella_Sky_Copenhagen tenant Marriott_International AC_Hotel_Bella_Sky_Copenhagen architect 3XN AC_Hotel_Bella_Sky_Copenhagen floorCount 23 """ Text:
Model output: |  The AC Hotel Bella Sky Copenhagen is owned by Bella Center, and is rented by Marriott International. It was designed by 3XN and has 23 floors.
Table 2:  Zero-Shot minimal instruction prompt. Template of the prompt and
complete example in English. Few-Shot in-context
---
Template: |  Write the following triples as fluent English | Irish | Welsh | Maltese | Breton text. Triple 1: """ {set of triples in the format subject predicate object and each triple in a new line} """ Text 1: {verbalisation of Triple 1} ## Triple 2: """ {set of triples in the format subject predicate object and each triple in a new line} """ Text 2: {verbalisation of Triple 2} ## Triple 3: """ {set of triples in the format subject predicate object and each triple in a new line} """ Text 3: [MODEL]
Example Prompt: |  Write the following triples as fluent English text. Triple 1: """ Adolfo_Suárez_Madrid–Barajas_Airport runwayName "14R/32L" """ Text 1: 14R/32L is the runway name of Adolfo Suárez Madrid-Barajas Airport. ## Triple 2: """ American_Journal_of_Mathematics abbreviation "Am. J. Math." American_Journal_of_Mathematics firstPublicationYear 1878 American_Journal_of_Mathematics issnNumber "1080-6377" """ Text 2: The American Journal of Mathematics was first published in 1878 and is also known by the abbreviated title of Am. J. Math. It has an ISSN number of 1080-6377. ## Triple 3: """ AC_Hotel_Bella_Sky_Copenhagen owner Bella_Center AC_Hotel_Bella_Sky_Copenhagen tenant Marriott_International AC_Hotel_Bella_Sky_Copenhagen architect 3XN AC_Hotel_Bella_Sky_Copenhagen floorCount 23 """ Text 3:
Model output: |  The AC Hotel Bella Sky Copenhagen is owned by Bella Center and is tenanted by Marriott International. It was designed by 3XN and has 23 floors.
Irish texts: |  Text 1: 14R/32L is ainm do rúidbhealach Aerfort Adolfo Suárez Madrid-Barajas Text 2: Foilsíodh an American Journal of Mathematics don chéad uair in 1878 agus aithnítear leis an ainm giorraithe Am. J. Math. chomh maith é. Tá an uimhir ISSN 1080-6377 aige.
Breton texts: |  Text 1: Anv leurenn bradañ aerborzh Adolfo Suárez Madrid-Barajas zo 14L/32R. Text 2: Finland zo bro ar Finniz hag hini ar skorndorrer Aleksey Chirikov bet savet e chanter-bigi Arctech en Helsinki.
Maltese and Welsh Triples: |  Triple 1: Albennie_Jones birthPlace Errata,_Mississippi Triple 2: GMA_New_Media industry Entertainment GMA_New_Media type Media_company GMA_New_Media product World_Wide_Web
Maltese texts: |  Text 1: Albennie Jones twieldet f’Errata Mississippi. Text 2: GMA New Media hija kumpanija tal-midja tal-industrija tad-divertiment li toffri servizzi li jikkonċernaw il-World Wide Web.
Welsh texts: |  Text 1: Ganed Albennie Jones yn Errata, Mississippi. Text 2: Mae GMA New Media yn gwmni cyfryngau yn y diwydiant adloniant sy’n cynnig gwasanaethau sy’n ymwneud â’r We Fyd Eang.
Table 3:  Few-Shot In-context prompt. Top Template and complete example in
English. Center Examples’ texts in Irish and Breton. Bottom Examples’ triple
set and texts in Maltese and Welsh.