# The complementary roles of non-verbal cues for
Robust Pronunciation Assessment

###### Abstract

Research on pronunciation assessment systems focuses on utilizing phonetic and
phonological aspects of non-native (L2) speech, often neglecting the rich
layer of information hidden within the non-verbal cues. In this study, we
proposed a novel pronunciation assessment framework, IntraVerbalPA. The
framework innovatively incorporates both fine-grained frame- and abstract
utterance-level non-verbal cues, alongside the conventional speech and phoneme
representations. Additionally, we introduce “Goodness of phonemic-duration”
metric to effectively model duration distribution within the framework. Our
results validate the effectiveness of the proposed IntraVerbalPA framework and
its individual components, yielding performance that either matches or
outperforms existing research works.

Index Terms—  Automatic pronunciation assessment, Non-verbal cues, End-to-End.

##  1 Introduction

Computer-assisted pronunciation training (CAPT) for foreign language learning
has seen a surge in global demand in recent years. CAPT benefits non-native
learners with personalized, cost-effective feedback, promotes self-directed
learning and improves pronunciation skills. It also offers flexibility
compared to traditional instruction [1, 2]. One of the main objective of the
CAPT is to automate pronunciation assessment (PA). To achieve this goal, the
automated PA model need to estimate a score that reflects the oral proficiency
based on some standardized assessment criteria [3, 1].

The task of PA is inherently subjective, even scores assigned by human expert
annotator often vary for the same spoken utterance. These discrepancies arises
from annotator’s unique experiences, their own interpretations of the scoring
guidelines, and/or their focus on specific aspect of pronunciation – like
fluency, prosody, word accuracy or even a combination. Hence, designing an
automated PA that emulate the annotators’ (or a teacher) is very much complex
and challenging. The challenges extends beyond the constraints of datasets
availability, and modeling intricacies, to include the crucial task of
selecting features and approach to model their representations.

Numerous investigations have explored a range of features and modeling
approaches aimed at enhancing modeling performance. These explorations have
encompassed the utilization of Goodness-of-Pronunciation (GOP) metrics [4, 5,
6], the integration of manually crafted handful of non-verbal features such as
duration, energy, and pitch [7, 8, 9], as well as the utilization of state-of-
the-art pre-trained self-supervised learning models for modeling improvement
[10, 11, 12].

Fig. 1: Overview of proposed IntraVerbalPA.

However, majority of the studies often neglect the rich layer of information
hidden within the non-verbal cues. For automated PA, integrating an additional
layer of non-verbal cues – pitch, intonation, voice quality, etc., can greatly
enhance the system’s ability to evaluate oral proficiency, bringing in the
human perception factor in the equation.

Therefore, we introduce an novel PA framework –

IntraVerbalPA. The framework is jointly trained to score the ‘fluency’ and
‘prosodic’ aspects in multi-task setup. IntraVerbalPA leverages both latent
speech and phoneme embedding, while complementing them with handcrafted frame-
and utterance-level non-verbal paralinguistic cues. We explored both
conventional and unconventional non-verbal cues – duration, pitch, jitter,
intonation, voice and unvoiced segment features among others, for modeling PA
scores. Furthermore, we introduce a novel metric to integrate the temporal
information in the model. The framework can seamlessly integrate any new
features for both frame-level and utterance-level, hence offering
opportunities to the researchers to investigate any new feature set. Our
contributions are as follows: (a) Integration of spoken non-verbal cues in the
PA system; (b) Inclusion of frame-level low-level descriptor in IntraVerbalPA;
(c) Utilization of utterance-level non-verbal functionals in IntraVerbalPA;
(d) Introduction of a new metric – Goodness of phonemic-duration, conditioned
on the duration distribution of native English speakers.

##  2 Proposed Framework

Figure 1 shows our proposed IntraVerbalPA framework, designed to train an
efficient end-to-end pronunciation assessment model using different sources of
information from the input signal. The IntraVerbalPA model comprised of $4$
modules, Speech Encoder, PhoneCue Encoder, Non-Verbal Features Encoder, and a
Projection Block.

##### Framework Overview

Given an input raw signal $\mathcal{X}$, of $n$ samples, we first extract
contextualized acoustic representations, $C_{t}$ (of dimension, $D:1024$),
from the Speech Encoder. Simultaneously, $\mathcal{X}$ is also passed through
the Non-Verbal Features Encoder to obtain non-verbal phoneme-level
($\mathbf{\tilde{F}_{nv}}$), utterance-level $\mathbf{U_{nv}}$ feature along
with duration, $GoPD$ representation. We then pass $\mathbf{\tilde{F}_{nv}}$
and $GoPD$ to the PhoneCue Encoder. The resultant output,
$\mathbf{\tilde{p}_{nv}}$, along with $C_{t}$, and $\mathbf{U_{nv}}$ are then
pass to the Projection Block for predicting Fluency and Prosodic scores.

###  2.1 Speech Encoder Module

The wav2vec2-large111https://huggingface.co/facebook/wav2vec2-large-robust
[13] model is a pre-trained wav2vec2.0 [14]. It follows the same architecture
as the wav2vec2.0 model. The encoder network consists of blocks of temporal
convolution layers with $512$ channels, and the convolutions in each block
have strides and kernel sizes that compress about $25$ms of $16$kHz audio
every $20$ms. The context network consists of $24$ blocks with model dimension
$1024$, inner dimension $4096$, and $16$ attention heads.

###  2.2 Non-Verbal Features Encoder

Inside the non-verbal feature encoder, using the input $\mathcal{X}$, we first
extract low level descriptors in frame-level ($\mathbf{F_{nv}}$) and apply
statistical functional to create utterance-level ($\mathbf{U_{nv}}$.)
representation using OpenSmile.222https://audeering.github.io/opensmile-
python/ We then align the input $\mathcal{X}$ with the canonical phoneme
sequence $\mathcal{Y}$ using the Alignment Module to convert frame-level non-
verbal $\mathbf{F_{nv}}$ representation to phoneme-level
($\mathbf{\tilde{F}_{nv}}$) representation. Moreover, we also use the phoneme-
level alignments to calculate the duration representation, $GoPD$.

####  2.2.1 Alignment Module

To align the canonical sequence with the audio, we opt for wav2vec2.0 trained
for frame-level classification
333https://huggingface.co/charsiu/en_w2v2_fc_10ms [15]. The phone alignment is
acquired through forced alignment performed using the Dynamic Time Warping
(DTW) algorithm based on models output probability matrix and the given
canonical phone transcription.

####  2.2.2 Goodness of phonemic-duration (GoPD)

We introduce a metric named Goodness of phonemic-duration (GoPD), inspired by
the goodness of pronunciation (GoP) metric [16], which is defined for a given
observation $\mathbf{O}$ and a phone $\mathbf{p}$ as follows:

$GOP(p)=P(p|O)=\frac{p(O|p)\ P(p)}{\sum_{q}\ p(O|q)\ P(q)}$ (1)

First, we extracted phoneme duration from native English (subset of TIMIT
[17]) data using the alignment module (in Section 2.2.1). We then construct
Gaussian distributions specific to each phoneme $\mathbf{p}$ denoted as
$\mathbf{D_{p}}$ to later use it in the IntraVerbalPA framework. Within the
framework, using the pre-extracted distribution, we compute the GoPD as
follows:

$GoPD(d_{t})=log(P_{D_{p_{t}}}(d_{t}))$ (2)

for a given duration $d_{t}$ corresponding to a L2-phoneme $p_{t}$.

In Figure 2, we present an illustration featuring two phonemes duration
distributions, ’V’ and ’OY’. Notably, ’OY’ exhibits a relatively higher mean
duration compared to ’V’, which aligns with our expectations since ’V’
represents a vowel sound and ’OY’ a consonant sound. However, it’s worth
noting that ’V’ displays a smaller standard deviation. This characteristic
makes ’V’ more sensitive to long duration, potentially signaling elongation.

Fig. 2: OY vs V duration distribution (ms).

####  2.2.3 Frame-level features

As shown in Table 1, we used Loudness, AlphaRatio, Pitch, and JitterLocal
extracted at frame-level using openSMILE[18]. The Energy is an important
feature of speech detection, the energy distribution may be related to the
intonation property, we presented Energy by Loudness, and AlphaRatio. Pitch
provides acoustic cues for a speaker’s intonation, confidence and
expressiveness, we present that using logarithmic F0, and JitterLocal.

Frame-level features |  Description |  Relevance
---|---|---
Loudness |  Estimate of perceived signal intensity from an auditory spectrum |  Intonation
AlphaRatio |  Ratio of the summed energy from 50-1000 Hz and 1-5 kHz. Represents the high-frequency content and the spectral balance. |  Intonation
Pitch |  logarithmic F0 on a semitone frequency scale |  Intonation, Confidence and Expressiveness
JitterLocal |  deviations in individual consecutive F0 period lengths |  Intonation, Confidence and Expressiveness

Table 1: Selected Frame-level features and their relevance

####  2.2.4 Utterance-level features

For utterance-level features, we selected Pitch features (mean, std,
percentiles, mean rising slope, std rising slope, mean falling slope, and std
falling slope) that provides information speaker’s intonation, and Voiced
segment features (mean, std), Unvoiced segment features (mean, std) that
provides information about disfluencies.

###  2.3 PhoneCue Encoder Module

The PhoneCue Encoder takes as input a sequence $Y={y_{1},y_{2},\ldots,y_{m}}$
representing parsed canonical phoneme sequence, then to an embedding layer
with dimension $D:41$. These embedding are projected using a feedforward
operation (with dimension $D:24$), resulting in the intermediate feature
vector $\mathbf{\tilde{p}_{nv}}$. Subsequently, we vertically concatenate this
intermediate feature vector $\mathbf{\tilde{p}{nv}}$ with other relevant
components, including GoPD and $\mathbf{\tilde{F}{nv}}$:

$\mathbf{r_{nv}}=\begin{bmatrix}GoPD\\\ \mathbf{\tilde{F}_{nv}}\\\
\mathbf{\tilde{p}_{nv}}\end{bmatrix}$ (3)

Finally, the $\mathbf{r_{nv}}$ is processed through a Bi-LSTM with dimension
($D:512$), resulting in the feature representation $\mathbf{p_{nv}}$
($D:1024$) capturing the non-verbal and phonetic cues present in the
utterance.

###  2.4 Projection Block

The $\mathbf{p_{nv}}$ and the contextualized acoustic representations $C_{t}$
are then passed to a attention layer that takes $\mathbf{p_{nv}}$ as key and
query, and $C_{t}$ as value, resulting in the final feature representation
$\mathbf{\alpha_{Cnv}}$ ($D:1024$)

$\mathbf{\alpha_{Cnv}}=Attention(K=\mathbf{p_{nv}},Q=\mathbf{p_{nv}},V=C_{t})$
(4)

The embeddings $C_{t}$ and $\alpha_{Cp}$ ($D:1024$) are then concatenated with
utterane-level features $\mathbf{U_{nv}}$, resulting in:

$F=C_{t}+\alpha_{Cpn}+\mathbf{U_{nv}}$ (5)

The resulting $F$ is then parsed to Bi-LSTM ($D:512$), and get concatenated
with the residual utterance-level features $\mathbf{U_{nv}}$ giving:
$\tilde{F}=BiLSTM(F)+\mathbf{U_{nv}}$. Following, $\tilde{F}$ is then passed
to two separate projection layers $Projection_{f}$, $Projection_{p}$ of
$D:11$44411: the dimension of scores ranging from 0-10, for respective Fluency
and Prosodic score classification.

##  3 Experimental Setup

###  3.1 Datasets

For the study, we used the widely used Speechocean762 [speechocean762] an
extensive dataset specifically designed for pronunciation assessment. The
dataset comprises a total of 5,000 English utterances obtained from 250 non-
native speakers. Each utterance in the dataset is associated with five aspect
scores at the utterance level, namely accuracy, fluency, completeness,
prosody, and a total score ranging from 0 to 10. To ensure reliability, each
of these scores is annotated by five expert evaluators. In our study, we are
interested in fluency and prosodic sentence level scores. $10\%$ of the
training data is used as the validation set for the model and early stopping
detection.

###  3.2 Model Training and Parameters

The models are optimized using Adam optimiser [19] for 50 epochs with early
stopping criterion ($=2$). The initial learning rate is set to $1\times
10^{-4}$, with a batch size of $32$. For all setups, the features extraction
layers remain frozen and only transformer layers are finetuned. The total loss
denoted as $loss_{total}$ criterion is the combination of CE loss on fluency
scores $CE_{f}$, and CE loss on prosodic scores $CE_{p}$ defined as:

$\small loss_{total}=0.5\times CE_{f}+0.5\times CE_{p}$ (6)

###  3.3 Evaluation

We reported Pearson Correlation Coefficient (PCC) between model prediction and
the average score of teachers provided in the dataset. To overcome any
randomness in the reported results, we repeated each experiment 3 times and
reported average PCC.

##  4 Results and Discussion

Table 2 demonstrate the efficacy of the proposed IntraVerbalPA in modeling the
Fluency and Prosodic scores.

Baselines: The proposed model significantly outperform the traditional
approach of modeling via fine-tuning a pre-trained model to the task, with and
without encoded canonical phoneme embedding (Table 2: Baseline).

Comparision to Prior Studies: In comparison to contemporary models (see Table
2: Contemporary), the IntraVerbalPA performs comparably with the MultiPA [9],
and Joint-CAPT [20]. Notably, while MultiPA and Joint-CAPT operate in a multi-
task setup context using either additional pre-trained features embeddings
such RoBERTA, using external L2-Artic [21] and TIMIT datasets, our
IntraVerbalPA focuses on monolingual settings and leverages a limited
L2-English Speechocean762 dataset.

Another model, 3M [8], integrates embeddings from three distinct self-
supervised learning models —- WavLM, HuBERT, and Wav2Vec2.0 (Large), along
with phoneme embeddings and additional features like transformer-based GOP,
duration, and energy. Despite its more comprehensive feature set and larger
number of parameters, IntraVerbalPA achieves a close results using a scaled-
down features.

Component Ablations: The enhanced performance of IntraVerbalPA reflects the
potential of non-verbal cues exploited in this work. To understand the
importance of the three main proposed components: (a) frame-level integration
of low-level non-verbal descriptor $\mathbf{\tilde{F}_{nv}}$, (b) utterance
level non-verbal features $\mathbf{U_{nv}}$, and (c) the Goodness of phonemic-
duration $GoPD$, we performed ablation tests. The PCC reported on Table
2:AblationComponents, reconfirmed the importance of integration fine-grained
frame-level non-verbal cues. Removing the $\mathbf{\tilde{F}_{nv}}$, drops the
PCC by $\approx 10\%$ for both fluency and prosodic scores. We observed the
duration representation, $GoPD$, also plays a key role in the model
performance, specially more for prosodic information than the fluency. This
aligns with previous findings showing the importance of modeling duration for
PA systems. When the $\mathbf{U_{nv}}$ is removed, the performance of the
IntraVerbalPA is less affected. However, the drop of absolute $1.12\%$ and
$2.16\%$ in prosodic and fluency scores respectively reflects the added
knowledge from $\mathbf{U_{nv}}$.

Age Group Ablations: PCC reported on Table 2:AblationAge suggest that
IntraVerbalPA performs exceptionally well for adult speech. For children’s
speech, the model shows satisfactory performance, although we lack prior
comparative data for a detailed analysis.

PCC | Prosodic | Fluency
---|---|---
Contemporary and Proposed Works
Raw Speech ($C_{t}$) [20] | 65.00% | 65.20%
Wav2vec-large [10] | 72.00% | 72.00%
GOPT [5] | 76.00% | 75.30%
Joint-CAPT-L1 [20] | 77.30% | 77.50%
Hubert-large-finetuned [10] | 77.00% | 78.00%
MultiPA [Multi-Task PA] [9] |  78.7% |  79.7%
3M [8] |  82.70% |  82.8%
IntraVerbalPA (Proposed) | 78.35% | 78.51%
Baseline Results
SSL:Wav2vec-large, ($Ct$) | 72.04% | 72.00%
$Ct$ \+ Phoneme Embedding | 70.4% | 70.94%
Ablation: Components
– GoPD (duration) | 70.14% | 71.60%
– $\mathbf{\tilde{F}_{nv}}$ (frame-level) | 69.26% | 69.22%
– $\mathbf{U_{nv}}$ (utt-level) | 77.23% | 76.35%
Ablation: Age Groups
IntraVerbalPA (Adult) |  85.31% |  85.47%
IntraVerbalPA (Children) |  57.95% |  59.80%

Table 2: Reported Pearson correlation coefficient, PCC, for the prior and
contemporary works; proposed IntraVerbalPA, along with baselines and ablation
of crucial components of the network. GoPD: Goodness of phonemic-duration,
$\mathbf{\tilde{F}_{nv}}$ converted phoneme-level representation of non-verbal
cues, $\mathbf{U_{nv}}$: utterance-level non-verbal cues representation.

##  5 Conclusion

We introduce the IntraVerbalPA framework, enriched with both fine-grained and
abstract non-verbal cues along with the conventional speech and phoneme
representation for modeling pronunciation assessment system. Moreover, we
propose a new metric to effectively model duration distribution within the
framework. Our reported results validate the importance of individual
components of the framework, and demonstrate the efficacy of the
IntraVerbalPA. The framework is designed to seamlessly integrate both frame-
level and utterance-level information, thereby offering researchers ample
opportunities to explore additional features for further investigation.

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difference $\operatorname{\mathbf{x}}-\operatorname{\mathbf{x}}^{\prime}$ is
contained in the lattice $\Lambda$.

∎

###### Completion of proof of Theorem 1.2 and Proposition 7.2.

We start with the case $0<\tau<1$. The constants ${\mathscr{I}}(w_{R}^{\pm})$
obtained from Theorem 2.3 can be replaced with ${\mathscr{I}}_{R}(1+O(\eta))$
by using Lemma 7.1. We conclude from Lemma 7.3 and Proposition 6.5 that for
$\eta\gg B^{-1/2}$ we have

$\displaystyle{\mathscr{N}}_{{\mathscr{W}}}((R,\boldsymbol{\xi}),(L,\boldsymbol{\Gamma});B)$
$\displaystyle=B^{n-1}{\mathscr{I}}_{R}\mathfrak{S}({\mathscr{W}};L,\boldsymbol{\Gamma})(1+O(\eta))+O_{\varepsilon,\tau}\left(\frac{B^{n-1+\varepsilon}}{L^{n}R^{n-1}}E_{\tau}(\eta;B)\right)$
$\displaystyle=B^{n-1}{\mathscr{I}}_{R}\mathfrak{S}({\mathscr{W}};L,\boldsymbol{\Gamma})\left(1+O(\eta)+O_{\varepsilon,\tau}(B^{\varepsilon}E_{\tau}(\eta;B))\right),$

where $E_{\tau}(\eta;B)$ is given by (2.11).

The error term above is minimised by choosing $\eta$ such that
$\eta=E_{\tau}(\eta;B)$. When $n\geqslant 5$, this gives
$\eta=B^{\frac{-(n-3)\tau}{5n-7}}$ , and when $n=4$, this gives
$\eta=B^{-2\tau/17}$. This provides the power-saving error terms claimed in
Theorem 1.2.

In the case $-1<\tau\leqslant 0$, by choosing $\eta\sim 1$, we obtain

${\mathscr{N}}_{{\mathscr{W}}}((R,\boldsymbol{\xi}),(L,\boldsymbol{\Gamma});B)\ll_{\tau}\frac{B^{n-1+\varepsilon}}{L^{n}R^{n-1}}E_{\tau}(1;B)+\frac{B^{\frac{(n+1)(1-\tau)}{2}+\varepsilon}}{L^{n+1}R^{n}},$

which completes the proof of Proposition 7.2. ∎

###  7.3. Proof of Theorem 1.1 – From the affine cone to its projective image

Since $\dim V\geqslant 3$, $\operatorname{rank}\operatorname{Pic}(V)=1$, and
so $W^{o}$ is the unique universal torsor of $V$ (up to isomorphism).

In the following we assume that $R\geqslant R_{0}$, where $R_{0}$ is defined
in (7.1). For every
$\boldsymbol{\Gamma}\in{\mathscr{W}}^{o}({\mathbb{Z}}/L{\mathbb{Z}})$, let us
define the counting function
${\mathscr{N}}_{{\mathscr{W}}^{o}}((R,\boldsymbol{\xi}),(L,\boldsymbol{\Gamma});B)$
for ${\mathscr{W}}^{o}$ with the local conditions
$(R,\boldsymbol{\xi}),(L,\boldsymbol{\Gamma})$ to be

$\displaystyle\sum_{\begin{subarray}{c}\boldsymbol{x}\in{\mathscr{W}}^{o}({\mathbb{Z}})\\\
\boldsymbol{x}\equiv\boldsymbol{\Gamma}\bmod
L\end{subarray}}1_{B,R}(\boldsymbol{x})$
$\displaystyle=\\#\\{\boldsymbol{x}\in{\mathbb{Z}}^{n+1}\setminus\boldsymbol{0}:F(\boldsymbol{x})=0,\pi(\boldsymbol{x})\in{\mathscr{U}}(R,\boldsymbol{\xi}),\boldsymbol{x}\equiv\boldsymbol{\Gamma}\bmod
L,\gcd_{0\leqslant i\leqslant
n}(x_{i})=1,\|\boldsymbol{x}\|_{\boldsymbol{\xi}}\leqslant B\\}.$

It is related to the corresponding counting function for ${\mathscr{W}}$ via a
Möbius inversion:

${\mathscr{N}}_{{\mathscr{W}}^{o}}((R,\boldsymbol{\xi}),(L,\boldsymbol{\Gamma});B)=\sum_{d\in{\mathbb{N}}}\mu(d){\mathscr{N}}_{{\mathscr{W}}}((R,\boldsymbol{\xi}),(L,\overline{d}\boldsymbol{\Gamma});B/d).$

where we observe that whenever $d\boldsymbol{x}\equiv\boldsymbol{\Gamma}\bmod
L$, we have $(d,L)\mid\boldsymbol{\Gamma}$. So the $d$-sum is restricted to
$(d,L)=1$, and for every such $d$, we write $\overline{d}$ for the
multiplicative inverse modulo $L$.

Now for every $p\mid L$, we write $m_{p}:=\operatorname{ord}_{p}(L)$ and
define

${\mathscr{E}}_{p}(L,\boldsymbol{\Gamma}):=\\{y_{p}\in{\mathscr{W}}^{o}({\mathbb{Z}}_{p}):y_{p}\equiv\boldsymbol{\Gamma}\bmod
p^{m_{p}}\\}\subseteq{\mathscr{W}}^{o}({\mathbb{Z}}_{p}),$ (7.9)

and

${\mathscr{E}}(L,\boldsymbol{\Gamma}):=\prod_{p\mid
L}{\mathscr{E}}_{p}(L,\boldsymbol{\Gamma})\times\prod_{p\nmid
L}{\mathscr{W}}^{o}({\mathbb{Z}}_{p})\subseteq\prod_{p<\infty}{\mathscr{W}}^{o}({\mathbb{Z}}_{p}),$

which is a non-empty finite adelic open set of ${\mathscr{W}}^{o}$.

Let us take
$\boldsymbol{\Gamma}\in{\mathscr{W}}^{o}({\mathbb{Z}}/L{\mathbb{Z}})$ such
that $\pi(\boldsymbol{\Gamma})\equiv\boldsymbol{\Lambda}\bmod L$. We then have

$\pi^{-1}({\mathscr{D}}(L,\boldsymbol{\Lambda}))=\bigsqcup_{\gamma\in{\mathbb{G}}_{\operatorname{m}}({\mathbb{Z}}/L{\mathbb{Z}})}{\mathscr{E}}(L,\gamma\boldsymbol{\Gamma}).$
(7.10)

By (7.10) we can now relate the counting functions of ${\mathscr{W}}^{o}$ and
${\mathscr{W}}$ to that of $V$ (1.2):

$\displaystyle{\mathscr{N}}_{V}((R,\boldsymbol{\xi}),(L,\boldsymbol{\Lambda});B)$
$\displaystyle=\frac{1}{2}\sum_{\gamma\in{\mathbb{G}}_{\operatorname{m}}({\mathbb{Z}}/L{\mathbb{Z}})}{\mathscr{N}}_{{\mathscr{W}}^{o}}((R,\boldsymbol{\xi}),(L,\gamma\boldsymbol{\Gamma});B)$
$\displaystyle=\frac{1}{2}\sum_{\gamma\in{\mathbb{G}}_{\operatorname{m}}({\mathbb{Z}}/L{\mathbb{Z}})}\sum_{\begin{subarray}{c}(d,L)=1\\\
d\ll
B\end{subarray}}\mu(d){\mathscr{N}}_{{\mathscr{W}}}((R,\boldsymbol{\xi}),(L,\gamma\overline{d}\boldsymbol{\Gamma});B/d).$

The lemma below allows to extend the $d$-sum to infinity.

######  Lemma 7.5.

We have

$\begin{split}{\mathscr{N}}_{V}((R,&\boldsymbol{\xi}),(L,\boldsymbol{\Lambda});B)\\\
&=B^{n-1}\frac{{\mathscr{I}}_{R}}{2}\sum_{\gamma\in{\mathbb{G}}_{\operatorname{m}}({\mathbb{Z}}/L{\mathbb{Z}})}\sum_{(d,L)=1}\frac{\mu(d)}{d^{n-1}}\mathfrak{S}({\mathscr{W}};L,\gamma\overline{d}\boldsymbol{\Gamma})+O_{\varepsilon}\left(\frac{B^{n-1+\varepsilon}}{L^{n-1}R^{n-1}}E_{\tau}(B)\right).\end{split}$

###### Proof.

We split the summation over $d$ into different ranges. For $1\leqslant d\ll
B$, we define real numbers $\tau_{d}$ such that

$(LR)^{2}\asymp\left(\frac{B}{d}\right)^{1-\tau_{d}}.$ (7.11)

Note that we can assume $\tau_{d}<1$ for all $d$ under consideration.
Furthermore, we have $\tau_{d}>\theta^{\prime}$ for some fixed small constant
$\theta^{\prime}>0$, provided that $d\ll B^{\tau-\varepsilon}$ for some
$\varepsilon>0$, and hence for these values of $d$ we are in a position to
apply Theorem 1.2. Note that we used again here that the dependence of the
implicit constants is continuous in $\tau$ in Theorem 1.2. Motivated by this,
we define $D_{1}=B^{\tau-\varepsilon}$ and we write

$\Sigma_{\leqslant
D_{1}}:=\frac{1}{2}\sum_{\gamma\in{\mathbb{G}}_{\operatorname{m}}({\mathbb{Z}}/L{\mathbb{Z}})}\sum_{\begin{subarray}{c}(d,L)=1\\\
d\leqslant
D_{1}\end{subarray}}\mu(d){\mathscr{N}}_{{\mathscr{W}}}((R,\boldsymbol{\xi}),(L,\gamma\overline{d}\boldsymbol{\Gamma});B/d).$

We also let $E_{\tau_{d}}(\cdot)$ be as defined in (1.4). With this notation
in hand, Theorem 1.2 implies that

$\displaystyle\Sigma_{\leqslant
D_{1}}=B^{n-1}\frac{{\mathscr{I}}_{R}}{2}\sum_{\gamma\in{\mathbb{G}}_{\operatorname{m}}({\mathbb{Z}}/L{\mathbb{Z}})}\sum_{\begin{subarray}{c}(d,L)=1\\\
d\leqslant
D_{1}\end{subarray}}\left(\frac{\mu(d)}{d^{n-1}}\mathfrak{S}({\mathscr{W}};L,\gamma\overline{d}\boldsymbol{\Gamma})(1+O_{\varepsilon}(B^{\varepsilon}E_{\tau_{d}}(B/d)))\right)$
$\displaystyle=B^{n-1}\frac{{\mathscr{I}}_{R}}{2}\sum_{\gamma\in{\mathbb{G}}_{\operatorname{m}}({\mathbb{Z}}/L{\mathbb{Z}})}\sum_{\begin{subarray}{c}(d,L)=1\\\
d\leqslant
D_{1}\end{subarray}}\frac{\mu(d)}{d^{n-1}}\mathfrak{S}({\mathscr{W}};L,\gamma\overline{d}\boldsymbol{\Gamma})+O_{\varepsilon}\left(\frac{B^{n-1+\varepsilon}}{(LR)^{n-1}}\sum_{d\leqslant
D_{1}}\frac{E_{\tau_{d}}(B/d)}{d^{n-1}}\right)$ (7.12)

In view of (7.11), the condition $(LR)^{2}\ll B^{1-\tau}$ may be alternatively
written as

$\left(\frac{B}{d}\right)^{-\tau_{d}}\ll dB^{-\tau}.$ (7.13)

Let $\theta=(n-3)/(5n-7)$ if $n\geqslant 5$ and $2/17$ if $n=4$, so that
$E_{\tau}(B)=B^{-\tau\theta}$. Using (7.13), we obtain

$\sum_{d\leqslant D_{1}}\frac{E_{\tau_{d}}(B/d)}{d^{n-1}}\ll\sum_{d\leqslant
D_{1}}\frac{(dB^{-\tau})^{\theta}}{d^{n-1}}\ll B^{-\tau\theta}\ll
E_{\tau}(B).$ (7.14)

In order to complete the summation over $d$, we note that

$\sum_{\gamma\in{\mathbb{G}}_{\operatorname{m}}({\mathbb{Z}}/L{\mathbb{Z}})}\sum_{\begin{subarray}{c}(d,L)=1\\\
d>D_{1}\end{subarray}}\frac{\mu(d)}{d^{n-1}}\mathfrak{S}({\mathscr{W}};L,\gamma\overline{d}\boldsymbol{\Gamma})\ll\frac{B^{n-1}}{(LR)^{n-1}}\sum_{d>B^{\tau-\varepsilon}}d^{1-n}\ll\frac{B^{n-1+\varepsilon}}{(LR)^{n-1}}B^{(2-n)\tau}.\\\
$ (7.15)

Inserting the estimates (7.14), (7.15) into (7.12), and observing that
$B^{(2-n)\tau}\ll E_{\tau}(B)$, we obtain

$\Sigma_{\leqslant
D_{1}}=B^{n-1}\frac{{\mathscr{I}}_{R}}{2}\sum_{\gamma\in{\mathbb{G}}_{\operatorname{m}}({\mathbb{Z}}/L{\mathbb{Z}})}\sum_{(d,L)=1}\frac{\mu(d)}{d^{n-1}}\mathfrak{S}({\mathscr{W}};L,\gamma\overline{d}\boldsymbol{\Gamma})+O_{\varepsilon}\left(\frac{B^{n-1+\varepsilon}}{L^{n-1}R^{n-1}}E_{\tau}(B)\right).$
(7.16)

Next, we let $D_{2}=B^{\frac{1+\tau}{2}-\varepsilon}$. Then for any
$d\leqslant D_{2}$, we have $\tau_{d}>-1+\theta^{\prime}$ for some positive
$\theta^{\prime}$, and so Proposition 7.2 applies. Therefore, the quantity

$\Sigma_{D_{1}<d\leqslant
D_{2}}:=\frac{1}{2}\sum_{\gamma\in{\mathbb{G}}_{\operatorname{m}}({\mathbb{Z}}/L{\mathbb{Z}})}\sum_{\begin{subarray}{c}(d,L)=1\\\
D_{1}<d\leqslant
D_{2}\end{subarray}}\mu(d){\mathscr{N}}_{{\mathscr{W}}}((R,\boldsymbol{\xi}),(L,\gamma\overline{d}\boldsymbol{\Gamma});B/d)$

satisfies the estimate

$\Sigma_{D_{1}<d\leqslant
D_{2}}\ll\sum_{\gamma\in{\mathbb{G}}_{\operatorname{m}}({\mathbb{Z}}/L{\mathbb{Z}})}\sum_{\begin{subarray}{c}(d,L)=1\\\
D_{1}<d\leqslant
D_{2}\end{subarray}}\left(\frac{(B/d)^{n-1+\varepsilon}}{L^{n}R^{n-1}}(E_{\tau_{d}}(1;B/d)+1)+\frac{(B/d)^{(n+1)(1-\tau_{d})/2+\varepsilon}}{L^{n+1}R^{n}}\right).$
(7.17)

First, we compute

$\displaystyle\sum_{\gamma\in{\mathbb{G}}_{\operatorname{m}}({\mathbb{Z}}/L{\mathbb{Z}})}\sum_{\begin{subarray}{c}(d,L)=1\\\
D_{1}<d\leqslant
D_{2}\end{subarray}}\frac{(B/d)^{(n+1)(1-\tau_{d})/2}}{L^{n+1}R^{n}}$
$\displaystyle\ll\sum_{D_{1}<d\leqslant
D_{2}}\frac{(B/d)^{(n+1)(1-\tau_{d})/2}}{(LR)^{n}}\ll\sum_{D_{1}<d\leqslant
D_{2}}\frac{(LR)^{n+1}}{(LR)^{n}}\ll D_{2}LR\ll
B^{\frac{1+\tau}{2}-\varepsilon}B^{\frac{1-\tau}{2}}\ll B.$

For the remaining terms in (7.17), very crude estimates suffice. We recall
that

$E_{\tau_{d}}(1;B/d)=\begin{cases}(B/d)^{-\frac{n-3}{2}\tau_{d}},&\textrm{ if
}n\geqslant 5;\\\ (B/d)^{-\tau_{d}}+(B/d)^{-\frac{1+\tau_{d}}{4}},&\textrm{ if
}n=4.\end{cases}$

Using (7.13), for $\tau_{d}<0$ and any $n\geqslant 4$, we have

$E_{\tau_{d}}(1;B/d)\ll(B/d)^{-\frac{n-2}{2}\tau_{d}}\ll(dB^{-\tau})^{\frac{n-2}{2}}\ll
d^{\frac{n-2}{2}}.$ (7.18)

On the other hand, if $\tau_{d}>0$, then clearly $E_{\tau_{d}}(1;B/d)\ll 1$,
and so the above estimate holds in any case. Plugging into (7.17), we find
that

$\displaystyle\sum_{\gamma\in{\mathbb{G}}_{\operatorname{m}}({\mathbb{Z}}/L{\mathbb{Z}})}\sum_{\begin{subarray}{c}(d,L)=1\\\
D_{1}<d\leqslant
D_{2}\end{subarray}}\frac{(B/d)^{n-1+\varepsilon}}{L^{n}R^{n-1}}(E_{\tau_{d}}(1;B/d)+1)$
$\displaystyle\ll\frac{B^{n-1}}{(LR)^{n-1}}\sum_{D_{1}<d\leqslant
D_{2}}\frac{d^{\frac{n-2}{2}}}{d^{n-1}}$
$\displaystyle\ll\frac{B^{n-1}}{(LR)^{n-1}}D_{1}^{1-\frac{n}{2}}.$

Since $D_{1}=B^{\tau-\varepsilon}$ and $n\geqslant 4$, it is clear that this
term is dominated by the error term already present in (7.16).

Finally, we consider the range $D_{2}<d\ll B$ and set

$\Sigma_{d>D_{2}}:=\frac{1}{2}\sum_{\gamma\in{\mathbb{G}}_{\operatorname{m}}({\mathbb{Z}}/L{\mathbb{Z}})}\sum_{\begin{subarray}{c}(d,L)=1\\\
d>D_{2}\end{subarray}}\mu(d){\mathscr{N}}_{{\mathscr{W}}}((R,\boldsymbol{\xi}),(L,\gamma\overline{d}\boldsymbol{\Gamma});B/d).$

For this we use Lemma 7.4 and find that

$\displaystyle\Sigma_{d>D_{2}}\ll
L\sum_{d>D_{2}}\left(1+\frac{\left(\frac{B}{d}\right)^{n+1}}{L^{n+1}R^{n}}+\sum_{k=1}^{n}\frac{\left(\frac{B}{d}\right)^{k}}{R^{k-1}L^{k}}\right)$
$\displaystyle\ll
L\sum_{d>D_{2}}\left(1+\frac{\left(\frac{B}{d}\right)^{n+1}}{L^{n+1}R^{n}}+\frac{\left(\frac{B}{d}\right)}{L}\right)$
$\displaystyle\ll BL+B\log B+\frac{B^{n+1}}{L^{n}R^{n}}D_{2}^{-n}.$

For $n\geqslant 4$, we have

$BL+B\log B\ll B^{1+\frac{1-\tau}{2}}\ll
B^{\left(\frac{1+\tau}{2}\right)(n-1)-2\tau}\ll\frac{B^{n-1-2\tau}}{(LR)^{n-1}}\ll\frac{B^{n-1}}{(LR)^{n-1}}E_{\tau}(B),$

and

$\frac{B^{n+1}}{L^{n}R^{n}}D_{2}^{-n}\ll\frac{B^{n-1}}{(LR)^{n-1}}\cdot\frac{B^{2}}{LR}\cdot\left(B^{\frac{1+\tau}{2}-\varepsilon}\right)^{-n}\ll\frac{B^{n-1+\varepsilon}}{(LR)^{n-1}}B^{2-\frac{n}{2}}\ll\frac{B^{n-1+\varepsilon}}{(LR)^{n-1}}E_{\tau}(B).$

Combining the estimates that we have obtained for three different ranges for
$d$, we conclude the proof of Lemma 7.5. ∎

###### Completion of proof of Theorem 1.1.

In view of (6.13) and the estimates in Proposition 6.5, it remains to analyse
arithmetic part of the main term in Lemma 7.5 and to show that it equals
$\omega^{V}_{f}\left({\mathscr{D}}(L,\boldsymbol{\Lambda})\right)$, the finite
Tamagawa measure of finite adelic neighbourhood
${\mathscr{D}}(L,\boldsymbol{\Lambda})$. As in the proof of Proposition 6.5,
we fix $L_{0}\in{\mathbb{N}}$ such that both ${\mathscr{V}}$ and
${\mathscr{W}}^{o}$ are regular modulo $L_{0}$, and upon decomposing the
neighbourhood ${\mathscr{D}}(L,\boldsymbol{\Lambda})$ into a disjoint union
whose level of congruence are all divisible by $L_{0}$, we may assume that
$L_{0}\mid L$.

We have

$\mathfrak{S}({\mathscr{W}};L,\gamma\overline{d}\boldsymbol{\Gamma})=\prod_{p\mid
L}\sigma_{p}({\mathscr{W}};L,\gamma\overline{d}\boldsymbol{\Gamma})\prod_{p\nmid
L}\sigma_{p}({\mathscr{W}}).$

For every $p\mid L$ (recalling (7.9),

$\displaystyle\sigma_{p}({\mathscr{W}};L,\gamma\overline{d}\boldsymbol{\Gamma})=\lim_{k\to\infty}\frac{\\#\left({\mathscr{W}}^{o}({\mathbb{Z}}/p^{k}{\mathbb{Z}})\cap{\mathscr{E}}_{p}(L,\gamma\overline{d}\boldsymbol{\Gamma})\right)}{p^{nk}}=\frac{1}{p^{nm_{p}}}.$

In particular its value is independent of $\gamma$ and $d$. Therefore we
obtain

$\begin{split}&\sum_{\gamma\in{\mathbb{G}}_{\operatorname{m}}({\mathbb{Z}}/L{\mathbb{Z}})}\sum_{(d,L)=1}\frac{\mu(d)}{d^{n-1}}\mathfrak{S}({\mathscr{W}};L,\gamma\overline{d}\boldsymbol{\Gamma})\\\
=&\prod_{p\mid
L}\left(1-\frac{1}{p}\right)\frac{1}{p^{(n-1)m_{p}}}\times\prod_{p\nmid
L}\left(1-\frac{1}{p^{n-1}}\right)\sigma_{p}({\mathscr{W}})\\\ =&\prod_{p\mid
L}\left(1-\frac{1}{p}\right)\sigma_{p}({\mathscr{V}};L,\boldsymbol{\Lambda})\times\prod_{p\nmid
L}\left(1-\frac{1}{p}\right)\sigma_{p}(V)=\omega^{V}_{f}\left({\mathscr{D}}(L,\boldsymbol{\Lambda})\right),\end{split}$

on using (6.14) (6.15) (6.16). This finishes the proof. ∎

##  Appendix A Rational approximations on quadrics

The goal of this appendix is to first recall the “$\alpha$-constant”
introduced in the recent work [19] of McKinnon-M. Roth, which pioneers the
study of Diophantine approximation on algebraic varieties. (For later
developments see also [14].) This is a generalisation of the classical notion
of “irrationality measure” originally attached to real numbers. We then
compute its value achieved on generic open subsets on projective quadrics, so
as to deduce the optimality of Theorem 1.1.

###  A.1. Approximation constants

To ease the exposition, let us place ourselves in the following setting. Let
$X\hookrightarrow{\mathbb{P}}^{n}$ be an irreducible projective algebraic
variety defined over ${\mathbb{Q}}$. Let $L$ be an ample line bundle, to which
we associate a height function $H_{L}$. We choose a projective distance
function $d_{\nu}$ on ${\mathbb{P}}^{n}$ for a place $\nu$ of ${\mathbb{Q}}$
as in [19, §2]. Let $Q_{0}\in X({\mathbb{Q}}_{\nu})$ be a fixed $\nu$-adic
point.

######  Definition A.1.

For any subvariety $Y\subset X$, we define the (best) approximation constant
$\alpha^{\nu}(Q_{0},Y)$ (depending also on $L$ and $\nu$) to be the infimum of
$\gamma>0$ such that there exist infinitely many points $P\in Y({\mathbb{Q}})$
such that

$d_{\nu}(P,Q_{0})^{\gamma}H_{L}(P)<1.$

It can be shown that (see [19, §2] for details) $\nu$-adically around $Q_{0}$
all distances are equivalent, hence the $\alpha$ constant does not depend on
the choice of $d_{\nu}$, nor on the choice of $H_{L}$, nor is the bound $1$ of
importance. The constant $\alpha^{\nu}(Q_{0},Y)$ measures how quickly the
height must grow for a sequence of rational points of $Y$ that tends to
$Q_{0}$. The smaller the $\alpha$-constant is, the slower the height is
allowed to grow, which means $Q_{0}$ is better approximable. In dimension one
the approximation constant for algebraic points is known, as a corollary of
the classical K. Roth theorem ([19, Theorem 2.16]) for ${\mathbb{P}}^{1}$.

We can furthermore ask for approximating the point $Q_{0}$ in a “generic
direction”. This motivates the definition of another constant as follows. It
was first formulated by Pagelot (unpublished) and further developed in the
first author’s Ph.D. thesis [14].

######  Definition A.2.

With the notation above, we define the essential constant of $Q_{0}$ to be

$\alpha_{\operatorname{ess}}^{\nu}(Q_{0})=\sup_{\begin{subarray}{c}Y\subset
X\end{subarray}}\alpha^{\nu}(Q_{0},Y),$

where $Y$ ranges over all open dense subvarieties of $X$.

###  A.2. The essential constants for quadrics

We go back to our initial setting. Let $F\in{\mathbb{Z}}[x_{0},\ldots,x_{n}]$
be an integral non-degenerate quadratic form. Let $V\subset{\mathbb{P}}^{n}$
be the projective quadric defined by $F=0$. We always assume that $V$ is
${\mathbb{Q}}$-isotropic, that is $V({\mathbb{Q}})\neq\varnothing$. We shall
use the naive height $H:{\mathbb{P}}^{n}({\mathbb{Q}})\to{\mathbb{R}}_{>0}$
corresponding to ${\mathscr{O}}(1)$ and the global sections given by the
coordinates $x_{0},\ldots,x_{n}$.

Theorem 1.1 has the following consequence on bounding the essential
$\alpha$-constant from above.

######  Theorem A.3.

Assume that $n\geqslant 4$ (or equivalently $\dim V\geqslant 3$). Then for
every place $\nu$ of ${\mathbb{Q}}$, and for every $Q_{0}\in
V({\mathbb{Q}}_{\nu})$, we have

$\alpha_{\operatorname{ess}}^{\nu}(Q_{0})\leqslant 2.$

###### Proof.

Fix some proper closed subvariety $Z\subseteq V$. Recall the real neighborhood
$U_{0}\subseteq\mathbb{R}^{n-1}$ and the neighborhood ${\mathscr{U}}(R,Q_{0})$
from (1.1), which is an open real neighbourhood of side-length $R^{-1}$ around
$Q_{0}$. Pick a box $U_{1}\subseteq U_{0}$ such that
$g_{Q_{0}}^{-1}(R^{-1}U_{1})$ is disjoint from $Z(\mathbb{R})$. Choose a
smooth compactly supported weight function
$w_{4}:\mathbb{R}^{n-1}\rightarrow\mathbb{R}_{\geqslant 0}$ with support
contained in $U_{1}$. Define similarly to equation (2.10) a weight function

$w_{B,R,U_{1}}^{\pm}(\boldsymbol{x}):=w_{1}^{\pm}\left(R\frac{t_{j}}{t_{0}},j\geqslant
2\right)w_{2}\left(\frac{R^{2}}{B^{2}}F(\boldsymbol{x})\right)w_{3}^{\pm}\left(\frac{t_{0}(\boldsymbol{x})}{B}\right)w_{4}\left(R\frac{t_{j}}{t_{0}},j\geqslant
2\right),$ (A.1)

where we added an additional factor of $w_{4}$. Note that Theorem 2.3 still
holds with the additional weight function $w_{4}$. For $B$ sufficiently large
and for $0<\tau<1$, we find that

${\mathscr{N}}_{{\mathscr{W}}}(w_{B,R,U_{1}}^{\pm};(L,\boldsymbol{\Gamma}))\gg\frac{B^{n-1}}{L^{n}R^{n-1}},$

whenever $LR=O(B^{\frac{1-\tau}{2}})$. In particular, this quantity is
positive, so we can find a point $P=P(B;R,L)$ which is counted by
${\mathscr{N}}_{{\mathscr{W}}}(w_{B,R,U_{1}}^{\pm};(L,\boldsymbol{\Gamma}))$,
and in particular $P\not\in Z({\mathbb{Q}})$.

Case $\nu={\mathbb{R}}$. We take $L=1$ and we omit $(L,\boldsymbol{\Lambda})$.
For any fixed $\tau\in(0,1)$, define $R=B^{\frac{1-\tau}{2}}$. Using the
coordinate system $\boldsymbol{t}$ as given in Lemma 2.1, the real zoom
condition determined by the weight functions $w_{1}^{\pm},w_{4}$ implies that

$d_{{\mathbb{R}}}(P,Q_{0})\ll\frac{1}{R}$

for any $P$ as above and for any fixed choice of real distance functions
around $Q_{0}$. Hence

$d_{{\mathbb{R}}}(P,Q_{0})^{\frac{2}{1-\tau}}H(P)\ll\frac{B}{R^{\frac{2}{1-\tau}}}=1,$

from which we infer

$\alpha^{{\mathbb{R}}}(Q_{0},V\setminus Z)\leqslant\frac{2}{1-\tau}.$

With this we find

$\alpha_{\operatorname{ess}}^{{\mathbb{R}}}(Q_{0})\leqslant\inf_{0<\tau<1}2(1-\tau)^{-1}=2.$

Case $\nu$ ultrametric. We similarly take $R=1$ and $L=p^{m_{p}}$ where $p$
corresponds to $\nu$. Now the $p$-adic condition determined by the pair
$(p^{m_{p}},Q_{0})$ implies that

$d_{\nu}(P,Q_{0})\ll p^{-m_{p}},$

for any $P$ as above and for any fixed $p$-adic distance functions around
$Q_{0}$. The arguments now follow similarly as in the archimedean case. ∎

We now give the exact value of the essential constant of any rational point on
$V$ via elementary arguments. This shows the optimality of Theorem A.3 (and
hence Theorem 1.1).

######  Proposition A.4.

Assume that $n\geqslant 2$. Then for any place $\nu$ and for any $Q_{0}\in
V({\mathbb{Q}})$, we have

$\alpha_{\operatorname{ess}}^{\nu}(Q_{0})=2.$

###### Proof.

We shall prove that

$\alpha^{\nu}(Q_{0},U)\geqslant 2,$ (A.2)

for an appropriately chosen open subset $U\subset V$, which directly implies
$\alpha_{\operatorname{ess}}^{\nu}(Q_{0})\geqslant 2$. In fact (A.2) is a
consequence of the following repulsion inequality

$\operatorname{d}_{\nu}(P,Q_{0})^{2}H(P)\geqslant A_{Q_{0}},$ (A.3)

valid for any $P\in U({\mathbb{Q}})$ different from $Q_{0}$, where
$A_{Q_{0}}>0$ depending only on $Q_{0}$. We are going to establish (A.3).

Upon a ${\mathbb{Q}}$-linear transformation as in Lemma 2.1, which clearly
does not affect the value of $\alpha$ (see [19, Proposition 2.4]), we may
assume that $Q_{0}=[1:0:\cdots:0]$ and

$F(x_{0},\ldots,x_{n})=x_{0}x_{1}+F_{2}(x_{2},\ldots,x_{n}).$

We take $\left(\frac{x_{1}}{x_{0}},\ldots,\frac{x_{n}}{x_{0}}\right)$ as a
local chart for the open set $W:=(x_{0}\neq 0)$. We may assume that
$\max_{1\leqslant i\leqslant n}|x_{i}/x_{0}|\ll 1$, else (A.3) is trivial. By
[19, Proposition 2.4] projective distance functions coming from different
embeddings are equivalent. So we may use the following projective distance
function

$\operatorname{d}_{\nu}(P,Q_{0})=\max_{1\leqslant i\leqslant
n}\left|\frac{x_{i}}{x_{0}}\right|_{\nu},$

whenever $P=[x_{0}:\cdots:x_{n}]\in W({\mathbb{Q}})$. (In what follows,
homogeneous coordinates of $P$ are all integers with
$\gcd(x_{0},\ldots,x_{n})=1$.) We define the closed subset
$V_{1}:=V\cap(x_{1}=0)$, and the open subset

$U:=(W\cap V)\setminus V_{1}.$ (A.4)

We first consider the case where $\nu$ is real. Since for every
$P=[x_{0}:\cdots:x_{n}]\in U({\mathbb{Q}})$,

$|x_{0}|\leqslant|x_{0}x_{1}|=|F_{2}(x_{2},\ldots,x_{n})|\ll\max_{2\leqslant
i\leqslant n}|x_{i}|^{2},$

we then have

$\operatorname{d}_{{\mathbb{R}}}(P,Q_{0})^{2}H(P)\geqslant\left(\frac{\max_{2\leqslant
i\leqslant
n}|x_{i}|}{|x_{0}|}\right)^{2}|x_{0}|\gg\frac{|x_{0}|}{|x_{0}|^{2}}|x_{0}|\geqslant
1.$

We next assume that $\nu$ corresponds to a prime $p$. For every
$P=[x_{0}:\cdots:x_{n}]\in U({\mathbb{Q}})\setminus\\{Q_{0}\\}$ with coprime
integers $x_{0},\ldots,x_{n}$, the equation
$x_{0}x_{1}=-F_{2}(x_{2},\ldots,x_{n})$ implies that

$\operatorname{ord}_{p}(x_{1})+\operatorname{ord}_{p}(x_{0})\geqslant
2\min_{2\leqslant i\leqslant n}\operatorname{ord}_{p}(x_{i}),$

whence

$|x_{0}x_{1}|_{\nu}\leqslant\max_{2\leqslant i\leqslant n}|x_{i}|^{2}_{\nu}.$
(A.5)

Fix $i_{0}\geqslant 2$ (depending on $P$) such that $|x_{i_{0}}|^{2}_{\nu}$
attains the maximum above. We necessarily have $x_{i_{0}}\neq 0$ since
otherwise we would have $x_{2}=\cdots=x_{n}=0$ and this would imply $x_{1}=0$,
contradicting to $P\not\in V_{1}({\mathbb{Q}})$.

Note that by equation (A.5) we find that
$|x_{1}|_{\nu}^{2}\leqslant\max_{2\leqslant i\leqslant n}|x_{i}|_{\nu}^{2}$.
We conclude that.

$\operatorname{d}_{\nu}(P,Q_{0})^{2}H(P)\geqslant\frac{|x_{i_{0}}|_{\nu}^{2}}{|x_{0}|_{\nu}^{2}}|x_{1}|\geqslant\frac{\max_{2\leqslant
i\leqslant n}|x_{i}|^{2}_{\nu}}{|x_{0}x_{1}|_{\nu}}\geqslant 1.$

In summary, we have proven that for every $P\in
U({\mathbb{Q}})\setminus\\{Q_{0}\\}$ and for every $\nu$,

$\operatorname{d}_{\nu}(P,Q_{0})^{2}H(P)\gg 1.$

This confirms the claim (A.3).

Finally we prove the reverse inequality

$\alpha_{\operatorname{ess}}^{\nu}(Q_{0})\leqslant 2,$ (A.6)

for arbitrary $\nu\in\operatorname{Val}({\mathbb{Q}})$ and $Q_{0}\in
V({\mathbb{Q}})$. We shall exhibit a family of ${\mathbb{Q}}$-curves through
$Q_{0}$ which are _very free_ , meaning that their union sweeping out a dense
subset of the quadric. The idea essentially follows from the fact that
quadrics containing at least one rational point are rational. We fix a
rational hyperplane $H\subset{\mathbb{P}}^{n}$ not containing $Q_{0}$. Then
any generic line through $Q_{0}$ intersects $H$ at a unique point. This
defines a $2$-to-$1$ projection map $\operatorname{pr}$ from an open subset of
$V$ to $H$. Now the Zariski closure of $\operatorname{pr}^{-1}(l)$, the
preimage of any generic line $l\subset H$ via $\operatorname{pr}$ in $V$, is a
conic through $Q_{0}$. Moving the line $l\subset H$ gives a family of smooth
conics through $Q_{0}$, each one having $\alpha$-constant equal to $2$. This
confirms the claim (A.6). ∎

######  Remarks A.5.

  1. (1)

We can show that $\alpha^{\nu}(Q_{0},V_{1})=1$, whenever $Q_{0}\in
V({\mathbb{Q}})$ and $V_{1}({\mathbb{Q}})\setminus\\{Q_{0}\\}\neq\varnothing$
(or equivalently $F_{2}$ is isotropic), where $V_{1}$ is defined as in the
proof of Proposition A.4. So $V_{1}$ is “locally accumulating” in the sense of
[14], which morally contains rational points that are “better approximable” to
$Q_{0}$.

  2. (2)

Rational points are usually “badly approximable” compared to irrational ones,
i.e., they have larger $\alpha$-constants. Recent work of de Saxcé [29] deals
with rational approximations of a fixed real algebraic point on a projective
quadric based on a version of Schmidt’s subspace theorem, and it is proved
that such a point has smaller $\alpha$-constant.

  3. (3)

Although not directly related to the results of this article, it is worth
pointing out that studying the distribution of rational points inside of a
shrinking neighbourhood of size corresponding exactly to the essential
constant, say $R\asymp B^{\frac{1}{2}}$, is an interesting (but challenging)
problem. The limit distribution is in general quite different from the one
given by the Tamagawa measures. See [14] and the references therein for more
details on this subject.111Two examples of quadratic surfaces (split or not
over ${\mathbb{Q}}$) were addressed in Pagelot’s thesis (unpublished). We are
able to determine the shape of limit measures for higher dimensional quadratic
hypersurfaces (unpublished).

## Acknowledgements

We thank Miriam Kaesberg for her effort in an earlier stage of this project.
We are grateful to Tim Browning for his interest. The hospitality of IST
Austria, Universität Göttingen, and Bayes Centre at ICMS contributes
substantially to the development of this paper. We thank these institutes
heartily. The third author was supported by the University of Bristol and the
Heilbronn Institute for Mathematical Research.

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