[affiliation=1,2]MiseulKim [affiliation=1]Soo-WhanChung [affiliation=1]YounaJi
[affiliation=2]Hong-GooKang [affiliation=1]Min-SeokChoi

# Speak in the Scene: Diffusion-based Acoustic Scene Transfer
toward Immersive Speech Generation

###### Abstract

This paper introduces a novel task in generative speech processing, Acoustic
Scene Transfer (AST), which aims to transfer acoustic scenes of speech signals
to diverse environments. AST promises an immersive experience in speech
perception by adapting the acoustic scene behind speech signals to desired
environments. We propose AST-LDM for the AST task, which generates speech
signals accompanied by the target acoustic scene of the reference prompt.
Specifically, AST-LDM is a latent diffusion model conditioned by CLAP
embeddings that describe target acoustic scenes in either audio or text
modalities. The contributions of this paper include introducing the AST task
and implementing its baseline model. For AST-LDM, we emphasize its core
framework, which is to preserve the input speech and generate audio
consistently with both the given speech and the target acoustic environment.
Experiments, including objective and subjective tests, validate the
feasibility and efficacy of our approach.

###### keywords:

audio/speech generation, acoustic scene transfer, latent diffusion

##  1 Introduction

In our daily lives, we encounter a plethora of environmental sounds or
acoustic scenes, including background noise, reverberation, and other auditory
stimuli such as chirping birds, laughing children, or the murmur of distant
conversations. The acoustic environment plays an essential role in human
interactions, infusing conversations with vitality and often serving as a
contextual cue for speech. For example, we often include supplementary sounds
when conversing with others or reading a book to improve comprehension of the
spoken content. Therefore, people often edit recorded speech with various
sound samples, creating an immersive experience akin to speech recorded in a
real environment.

Remarkable advances in the field of generative models enable the generation of
high-quality speech and audio. Early generative methods, such as variational
autoencoders (VAE) [1, 2] and generative adversarial networks (GAN) [3, 4],
have successfully produced natural-sounding speech signals, surpassing the
capabilities of previous discriminative methods. However, they often struggle
with instability during training, which can lead to unpredictable and
unsatisfactory outputs. Recent diffusion-based models have effectively
addressed these challenges, markedly raising generation performance and
achieving remarkable success in the extensive domains of audio and speech.
Some studies [5, 6, 7] have underscored its efficacy in modeling the
distribution of speech signals, especially in facilitating multi-speaker
speech synthesis as well as high-quality synthesis. In the realm of music
generation, diffusion has demonstrated its proficiency in crafting musical
notes across various instruments such as piano, guitar, beats, and even vocals
[8, 9, 10]. Furthermore, diffusion models can even generate general sounds by
learning auditory characteristics from text description prompts and producing
sound effects and foley sounds [11, 12, 13, 14]. Notably, AudioLDM [15, 16]
demonstrates exceptional performance in audio generation, showcasing its
ability to generate audio signals based on CLAP model [17, 18], retrieving
acoustic characteristics from reference audio or text captions. VoiceLDM [19],
a text-to-speech model, is built upon the framework of AudioLDM, and it
generates speech signals with prompts and transcriptions to control speech,
voice and various speaking environments. Similarly, DiffRENT [20] aims to
transfer acoustic characteristics such as ambient noise and reverberation from
reference speech to input speech signals.

Figure 1: Overview of the proposed model, AST-LDM. Dotted lines represent the
training phase, while solid lines denote the inference process. Blocks marked
with  indicate pre-trained components that are not updated during training.

In this paper, motivated by recent developments, we propose a novel speech
processing task, Acoustic Scene Transfer (AST), aimed at transferring the
target acoustic scene onto an input speech signal. Although the framework of
AST is similar to the foundational knowledge presented in DiffRENT, we further
generalize the paradigm to encompass general audio rather than focusing solely
on ambient noise. To the best of our knowledge, this is the first time such a
task has been proposed for immersive speech generation, and we expect that its
potential benefits across various speech processing fields. While this task
may appear similar to adding background sound or room impulse responses as
speech augmentation, AST does not simply replicate the acoustic scene of
reference signals. Instead, it re-generates acoustic scenes, allowing for
flexibility and diversity on generated scenes and relieving limitations
imposed by noise or room impulse response in datasets. Therefore, AST would
encourage more impactful and immersive speech generation beyond conventional
natural voice generation.

We implement a latent diffusion model (LDM) for AST, Acoustic Scene Transfer
Latent Diffusion Model (AST-LDM), which predicts latent embeddings of a pre-
trained variational autoencoder, following AudioLDM and VoiceLDM. Our proposed
model mimics the background sound and acoustic characteristics of the target
environment while retaining the spoken term and speaker identity present in
the input speech. AST-LDM not only transfers acoustic backgrounds but also
facilitates transitions from anechoic to noisy or reverberant environments,
and vice versa. Furthermore, we demonstrate that AST-LDM can be conditioned
not only by reference audios, as in DiffRENT, but also by text descriptions of
acoustic scenes with a help of CLAP embeddings. By leveraging text prompts,
the proposed model gains larger diversity and enriched representations of
acoustic scenes, enabling it to generate unseen acoustic situations.

##  2 Diffusion-based Audio Generation

Diffusion [21, 22] leads the field of generative learning by predicting the
data distribution from Gaussian noise, and it allows the model to generate
outputs with a large variability. The advent of latent diffusion [23], which
operates in a latent space, has reduced excessive computational costs for
processing raw data and increased the applicability of diffusion framework on
various domains. In audio generation, diffusion models have achieved
significant breakthroughs, as demonstrated in their capability in vision
modality. Its application in speech synthesis led to the high-quality
generation of speech waveforms conditioned by mel-spectrograms [24]. Beyond
speech synthesis, diffusion techniques have been applied in speech enhancement
[25, 26] and separation [27, 28], enabling the generation of clear speech from
noisy inputs. Moreover, diffusion has demonstrated its versatility in creating
a broad spectrum of audio content including natural sounds and instrumental
music.

The method most closely aligned with ours is AudioLDM, which generates audio
sounds by utilizing CLAP model to extract embeddings for the conditional
generation. It adopts a self-supervised training approach; the model is
trained to predict pre-trained latent embeddings for a broad spectrum of audio
sounds, with the CLAP embedding of the audio itself serving as a reference
prompt. During inference, the model employs the text encoder of CLAP model,
using text descriptions of sound to steer the generation process rather than
relying on the audio prompt. It is enabled by the common latent space between
the audio and text encoders of CLAP model, which benefits from its training in
cross-modal self-supervision. This approach is particularly effective in
generating a varied range of audio outputs, including the ability to generate
unseen audio outputs that are not present during the training stage. Inspired
by AudioLDM, VoiceLDM [19] focuses on text-to-speech (TTS) synthesis,
incorporating both text transcriptions and audio signals as conditions for
training. It not only focuses on synthesizing clean speech but also simulates
speech within the described environment. Leveraging CLAP model, VoiceLDM
adjusts the acoustic scene based on textual scene descriptions, employing a
dual classifier-free guidance method [29] to balance the influence of both
conditions. DiffRENT [20] presents another innovative method, focusing on the
transfer of acoustic scenes from reference to target speech. Based on a
standard diffusion model, it directly estimates the target mel-spectrogram,
employing an environmental encoder instead of CLAP model. The environmental
encoder is jointly trained with the diffusion model, sharing the same training
criteria. DiffRENT changes the ambient noise and room acoustics of the target
speech to match those of the reference while preserving the spoken content.

In this work, we draw upon the training and conditioning strategies as well as
frameworks of these models to generate optimal latent embeddings for AST. Our
work aims to extend the boundaries of audio generation, leveraging the
capabilities of diffusion models to create immersive speech experiences.

##  3 AST-LDM

In this section, we introduce our proposed model, AST-LDM, and outline its
training strategy for the acoustic scene transfer task. The overall structure
of AST-LDM is given in Figure 1. AST-LDM is conditioned on two prompts: the
reference prompt, which consists of the target environments, and the content
prompt, representing the input speech signal aimed at preserving the speaking
contents in the output signal. Our model is inspired by the structure of
AudioLDM which includes the diffusion model, CLAP encoder, VAE, and HiFi-GAN
[4]. The VAE is charged with extracting the target latent embedding from the
mel-spectrogram, while HiFi-GAN serves as the vocoder, converting the mel-
spectrogram into an audio waveform. In the following subsections, we will
offer a detailed demonstration of the modules of our proposed model.

###  3.1 Reference Prompt

In training stage, the reference prompt comprises speech and its speaking
environments, including background sounds and reverberation. The audio encoder
of CLAP model processes the reference prompt to obtain the representation of
acoustic scenes. Previous studies on AudioLDM and VoiceLDM have demonstrated
the capability of generating audio using CLAP embeddings, suggesting their
effectiveness in capturing both the acoustic environments and speaking styles
of the audio prompts. In the inference stage, we utilize text descriptions of
the target acoustic scenes in conjunction with the text encoder of CLAP model,
in addition to the reference audio prompt used during training. Since CLAP
model is pre-trained using cross-modal self-supervision, we expect that the
text description can effectively replace the audio prompt. Utilizing a text
prompt offers the potential for greater diversity and enriched representations
of acoustic scenes, enabling the model to generate scenarios that have not
been encountered before. In our experiments, we simulate reference prompts as
noisy, reverberant, or both noisy and reverberant scenes, and also clean
speech signals to be transferred into a clean environment.

###  3.2 Content Prompt

The content prompt consists of the target spoken term that is intended to be
retained in outputs of AST-LDM. Since our goal is to transfer acoustic scenes
across diverse environments, the content prompt may contain environmental
sounds behind speech, and these acoustic elements should not remain in output
signals. Consequently, it is essential to precisely preserve the speech signal
from the content prompt and condition it onto the diffusion model. To achieve
this, we designed a content encoder based on a transformer structure,
consisting of two cascaded modules. The content prompt goes into the filter
module, which returns a ratio mask using a sigmoid function to be multiplied
by the mel-spectrogram of the content prompt. The filter module removes
undesired acoustic components, ensuring stable conditioning. Subsequently, the
masked mel-spectrogram is passed to the embedding module to encode the content
prompts onto the latent space. Like the reference speech, we simulate the
content prompt using noisy, reverberant, both noisy and reverberant, and clean
speech signals.

###  3.3 Latent Diffusion

Given the reference and content embeddings as conditions, the latent diffusion
model is trained to predict the target audio embedding in the latent domain.
As mentioned, the latent embedding of the target audio is extracted using the
pre-trained VAE model, which compresses mel-spectrograms into stochastic
representations. We utilized the same U-Net structure used in AudioLDM for the
diffusion model. The reference embedding is repeated along the time axis and
concatenated with the input of the diffusion model, while the content
embedding is conditioned using the cross-attention method on the bottleneck
feature of the U-Net. The target audio is simulated by utilizing the speech
signal of the content prompt and the acoustic scenes of the reference prompt.
During the training stage, we used the same speech signal in both reference
and content prompts but simulated with different acoustic scenes, and the
simulated reference prompt also served as the target audio signal.

##  4 Experimental Settings

#### Data Preparation

For training, we utilized 960 hours of speech data from LibriSpeech [30]
corpus and curated additional audio samples from AudioCaps [31], a balanced
subset of AudioSet [32], and VGGSound [33] to provide background sounds. Room
acoustics were simulated using room impulse responses (RIRs) from DNS-
Challenge dataset [34], consisting of both simulated and real-recorded
samples. We divided the RIR dataset into two subsets: two simulated room
impulse responses from each room size and RWCP real impulse responses for
evaluation, with the remaining data allocated for training.

For evaluation, we employed both audio samples and text descriptions for the
reference prompt. Audio samples were obtained from the test-clean subset of
the LibriSpeech corpus for speech utterances, the eval set of AudioSet, and
the RIR evaluation set of DNS-Challenge dataset. We prepared 1,800 speech
samples from LibriSpeech, half for clean content and the rest for clean
reference prompts. Since we used same speech signal for reference and content
prompts during training, we aligned the genders of speakers within reference
prompts and content prompts. We overlaid acoustic scenes onto the clean speech
samples to create content and reference prompts. In the case of using text
descriptions for reference prompts, we generated 1,200 sentences111Samples
available: https://ast-ldm.github.io/demo using GPT4 [35], each describing an
acoustic scene with the format: `A [male/female] speaks in [place] with
[background sounds] behind.' For noiseless scenes, we used the text prompt: `A
[male/female] speaks in a quiet room'.

In addition, to compare with AudioLDM and VoiceLDM, we leveraged audio samples
of LibriSpeech and AudioCaps. Since AudioCaps involves text captions of audio
samples, we used them as a reference text prompt, as well as a reference audio
prompt, and transcriptions of LibriSpeech, were used for the text input of
VoiceLDM while we used corresponding speech signals for AST-LDM. We also
included real-world recordings from VCTK-DDS dataset, featuring clean-noisy
speech pairs captured in real environments with ambient noise and
reverberation. For VCTK-DDS dataset [36], we retained audio samples without
adding additional background sounds or reverberation, using them as target
audio examples. Following the evaluation data ontology of DiffRENT, we
selected 2 speakers (p234, p241) and a recording environment, livingroom1 in
Uber. To ensure consistency across datasets, audio samples were resampled to
16kHz and segmented into 10-second chunks, with shorter samples padded to fit
the duration. For the simulation, we mixed speech and environmental sound with
a random signal-to-noise ratio (SNR) ranging from 4 to 20 dB.

#### Model Structure

The architectural configurations of the diffusion model, CLAP model, VAE
model, and HiFi-GAN remain consistent with those described in [15, 19]. For
the content encoder, we designed it with 2 transformer layers for the filter
module and 2 convolution layers followed by 4 transformer layers for the
embedding module. Each transformer layer utilizes 256 hidden dimensions with 8
attention heads. Reference prompts, content prompts, and target speech signals
are transformed into 64-dimensional mel-spectrograms on a logarithmic scale,
extracted every 10ms with a frame length of 64ms.

#### Training Strategy

We adopt a training scheme similar to that of AudioLDM and VoiceLDM and
introduce the dual classifier-free guidance method [29] to train our model
conditioned by two different prompts. During training, we randomly dropped
reference and content embeddings with probabilities of 0.1 and 0.1,
respectively. The model is trained using L2 loss to estimate added noise
during the diffusion forward process. While the VAE, CLAP, and HiFi-GAN are
pre-trained and left frozen, the diffusion model and content encoder are
jointly trained.

#### Evaluation Protocols

During inference, we used a DDIM sampler with 100 sampling steps, setting 1.0
for both dual classifier-free guidance strengths. We measured performance in
terms of scene similarity and content preservation. Scene similarity was
assessed using CLAP-based Fréchet Audio Distance (FAD) and CLAP similarity
metrics. CLAP similarity was calculated using both the audio encoder (CLAPA)
and, if available, the text encoder (CLAPT). Content preservation was
evaluated using word-error-rate (WER) employing Whisper-small.en [37] model.
Also, speaker similarity (SSM) between content prompts and output signals was
measured using the embedding distance of RawNet3 [38] model. For the
subjective evaluation, we conducted listening tests with 17 listeners to
assess the overall perceptual quality of the generated audio samples. The
assessment of audio quality was conducted using a 5-point mean-opinion-score
(MOS) scale, where scores ranged from 1 (bad) to 5 (excellent). Lower scores
for FAD and WER indicate better performance, while higher scores for CLAP, SSM
and MOS suggest better performance.

Table 1: Objective evaluation of generated speech with simulated dataset conditioned on audio reference prompts ($\mathcal{R}$: $\mathcal{A}$). Scenario ($\mathcal{R}:\mathcal{A}$) | FAD | CLAPA |  WER (%) | SSM
---|---|---|---|---
Clean $\rightarrow$ Clean | 0.345 | 0.633 | 4.8 | 0.814
Clean $\rightarrow$ Env | 0.278 | 0.627 | 5.8 | 0.785
Env $\rightarrow$ Clean | 0.554 | 0.479 | 22.6 | 0.718
Env $\rightarrow$ Env | 0.276 | 0.609 | 24.0 | 0.706
Table 2: Objective evaluation of generated speech with simulated dataset conditioned on text reference prompts ($\mathcal{R}$: $\mathcal{T}$). Scenario ($\mathcal{R}:\mathcal{T}$) | FAD | CLAPT |  WER (%) | SSM
---|---|---|---|---
Clean $\rightarrow$ Clean | 1.096 | 0.445 | 6.8 | 0.779
Clean $\rightarrow$ Env | 0.681 | 0.463 | 7.7 | 0.701
Env $\rightarrow$ Clean | 1.114 | 0.437 | 31.3 | 0.654
Env $\rightarrow$ Env | 0.690 | 0.446 | 25.7 | 0.673
Table 3: Comparison of baselines and the proposed model. $\mathcal{R}$ denotes
the reference prompt.

Model | $\mathcal{R}$ | FAD | CLAPA | CLAPT |  WER (%)
---|---|---|---|---|---
AudioLDM | $\mathcal{A}$ | 0.298 | 0.600 | 0.349 | -
| $\mathcal{T}$ | 0.697 | 0.359 | 0.349 | -
| Mean | 0.498 | 0.480 | 0.349 | -
VoiceLDM | $\mathcal{A}$ | 0.288 | 0.643 | 0.376 | 14.3
| $\mathcal{T}$ | 0.753 | 0.417 | 0.361 | 12.2
| Mean | 0.521 | 0.530 | 0.369 | 13.3
AST-LDM | $\mathcal{A}$ | 0.323 | 0.591 | 0.408 | 8.6
| $\mathcal{T}$ | 0.602 | 0.479 | 0.447 | 11.4
| Mean | 0.463 | 0.535 | 0.428 | 10.0

##  5 Experiment Results

In this section, we present the results of our experiments across different
datasets and metrics. Demo audio samples from our experiments are available
online11footnotemark: 1.

###  5.1 Performance on Simulated Recordings

In Table 1 and 2, we report the acoustic scene transfer performances of AST-
LDM across diverse pairs of content and reference prompts. Table 1 shows AST
results using audio signals as reference prompts ($\mathcal{R}:\mathcal{A}$),
while Table 2 utilizes text descriptions generated using GPT4 for the prompt
($\mathcal{R}:\mathcal{T}$). Each table showcases 4 scenarios for transferring
acoustic scenes: Clean $\rightarrow$ Clean, Clean $\rightarrow$ Env, Env
$\rightarrow$ Clean, and Env $\rightarrow$ Env. Here, `Env' involves
environmental scenes overlaid on speech.

Our findings in Table 1 demonstrate that AST-LDM effectively transfers
acoustic scenes while maintaining spoken contents and their speakers. However,
AST-LDM appears to be particularly susceptible to the `Env $\rightarrow$
Clean' transition. This vulnerability arises from the challenge of describing
a `clean' scene, where CLAP model may not be sufficiently trained on `silence'
or `quiet' acoustic environments. In Table 2, we found similar results to
those obtained using audio reference prompts. Since the generated
transcriptions contain complex descriptions and diverse unseen sounding
objects, the results show low overall scores compared to Table 1. While it
still preserves the spoken term of content prompts well, it encounters
difficulties in generating the same environment described in text prompts. We
anticipate that these issues in each table can be addressed by fine-tuning
CLAP on such clean environmental datasets and utilizing more diverse text
descriptions. Additionally, providing suitable instructions or prompts to
generate desirable signals through text descriptions is important.

Table 4: Evaluation of generated speech with DDS dataset with environment reference prompts. $\mathcal{C}$ denotes the content prompt. Ground-truth MOS score is 4.206. Model | $\mathcal{C}$ | FAD | CLAPA | SSM | MOS
---|---|---|---|---|---
DiffRENT | Clean | 0.332 | 0.852 | 0.746 | 4.000
| Env | 0.457 | 0.792 | 0.779 | 4.235
AST-LDM | Clean | 0.358 | 0.842 | 0.723 | 3.676
| Env | 0.478 | 0.817 | 0.772 | 4.294

###  5.2 Performance on Real Recordings

In Table 3, we conducted a comparison between AST-LDM and CLAP-based baseline
models, utilizing both audio $\mathcal{A}$ and text captions $\mathcal{T}$
from the AudioCaps dataset as reference prompts $\mathcal{R}$. Based on the
FAD and CLAP scores, the proposed model demonstrates competitive generation
performance compared to the baseline models. Specifically, the signals
generated by AST-LDM exhibit a strong correlation with the textual
descriptions (CLAPT) of the desired acoustic scenes. When we measured CLAP
scores between audio and text captions of the AudioCaps dataset, they scored
0.521, and the overall CLAP score of our model is the closest to the score.
Additionally, there appears to be less performance degradation when using text
reference prompts compared to the baseline models. Moreover, our model
achieves better WER scores in both types of reference prompts, indicating that
AST-LDM effectively preserves the spoken term and accurately conveys it in the
output signal. These findings suggest that AST-LDM is capable of generating
speech signals accurately within a desired acoustic environment, making it
well-suited for the acoustic scene transfer task.

Table 4 presents the evaluation results on DDS datasets compared with DiffRENT
model, using the audio samples obtained from its demo page. While both models
aim to achieve the same task, DiffRENT is trained exclusively on DDS datasets,
which results in a better fit to the evaluation dataset. However, our proposed
model still demonstrates competitive performance, and in some metrics, it even
outperforms DiffRENT. Besides, AST-LDM has superiority to learn more various
audio characteristics using CLAP encoders while DiffRENT is effective in
describing simple environments such as ambient noise or reverberation relying
on the environmental encoder. Also, with the help of CLAP model, AST-LDM has
advantages in describing in ordinary language instruction while DiffRENT
always require the target audio sample.

##  6 Conclusion

In this study, we introduced a novel task, acoustic scene transfer (AST),
aimed at transforming the acoustic environments of input speech into target
acoustic scenes. We proposed the first AST task-oriented model, AST-LDM, based
on the latent diffusion model. By leveraging CLAP model for acoustic scene
analysis, AST-LDM successfully altered and represented environments within
input speech signals, effectively removing original acoustic scenes.
Experimental results have confirmed the potential of AST-LDM for the AST task,
demonstrating its ability to transfer acoustic scenes as intended while
preserving spoken terms and voice identity in the input speech.

We believe this research would contribute significantly to the field of speech
processing, offering enriched and immersive speech generation capabilities.
For future work, we recommend focusing on more precise model implementations
and refining training strategies. Additionally, we need to explore optimal
prompt provisions for generalization and enable wider and more nuanced
representations of reference prompts.

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###### Abstract

We improve the range of $\ell^{p}(\mathbb{Z}^{d})$-boundedness of the integral
$k$-spherical maximal functions introduced by Magyar. The previously best
known bounds for the full $k$-spherical maximal function require the dimension
$d$ to grow at least cubically with the degree $k$. Combining ideas from our
prior work with recent advances in the theory of Weyl sums by Bourgain,
Demeter, and Guth and by Wooley, we reduce this cubic bound to a quadratic
one. As an application, we improve upon bounds in the authors’ previous work
[1] on the ergodic Waring–Goldbach problem, which is the analogous problem of
$\ell^{p}(\mathbb{Z}^{d})$-boundedness of the $k$-spherical maximal functions
whose coordinates are restricted to prime values rather than integer values.

title = Improved $\ell^{p}$-Boundedness for Integral $k$-Spherical Maximal
Functions, author = Theresa C. Anderson, Brian Cook, Kevin Hughes, and Angel
Kumchev, plaintextauthor = Theresa C. Anderson, Brian Cook, Kevin Hughes, and
Angel Kumchev, plaintexttitle = Improved lp-Boundedness for Integral
k-Spherical Maximal Functions, runningtitle = Improved Bounds for Integral
Maximal Functions, runningauthor = T. C. Anderson, B. Cook, K. Hughes, and A.
Kumchev, keywords = Maximal functions, integral averages, surface measures,
Fourier transforms, circle method, exponential sums. year=2018, number=10,
received=2 August 2017, published=29 May 2018, doi=10.19086/da.3675,

[classification=text]

##  1 Introduction

Our interest lies in proving $\ell^{p}(\mathbb{Z}^{d})$-bounds for the
integral $k$-spherical maximal functions when $k\geq 3$. These maximal
functions are defined in terms of their associated averages, which we now
describe. Define a positive definite function $\mathfrak{f}$ on
$\mathbb{R}^{d}$ by

$\mathfrak{f}(\mathbf{x})=\mathfrak{f}_{d,k}(\mathbf{x}):=|x_{1}|^{k}+\dots+|x_{d}|^{k},$

and note that when $\mathbf{x}\in\mathbb{R}_{+}^{d}$,
$\mathfrak{f}(\mathbf{x})$ is the diagonal form $x_{1}^{k}+\dots+x_{d}^{k}$.
For $\lambdaup\in\mathbb{N}$, let $R(\lambdaup)$ denote the number of integral
solutions to the equation

$\mathfrak{f}_{d,k}(\mathbf{x})=\lambdaup.$ (1.1)

When $R(\lambdaup)>0$, define the normalized arithmetic surface measure

$\sigma_{\lambdaup}({\mathbf{x}}):=\frac{1}{R(\lambdaup)}{\mathbf{1}}_{\\{{\mathbf{y}}\in\mathbb{Z}^{d}:\mathfrak{f}(\mathbf{y})=\lambdaup\\}}({\mathbf{x}}).$

We are interested in averages given by convolution with these measures:

$\sigma_{\lambdaup}*f(\mathbf{x})=\frac{1}{R(\lambdaup)}\sum_{\mathfrak{f}(\mathbf{y})=\lambdaup}f(\mathbf{x}-\mathbf{y})$

for functions $f:\mathbb{Z}^{d}\to\mathbb{C}$.

We know from the literature on Waring’s problem that as $\lambdaup\to\infty$,
one has the asymptotic

$R(\lambdaup)\sim\mathfrak{S}_{d,k}(\lambdaup)\lambdaup^{{d/k-1}},$ (1.2)

where $\mathfrak{S}_{d,k}(\lambdaup)$ is a convergent product of local
densities:

$\mathfrak{S}_{d,k}(\lambdaup)=\prod_{p\leq\infty}\muup_{p}(\lambdaup).$

Here $\muup_{p}(\lambdaup)$ with $p<\infty$ is related to the solubility of
(1.1) over the $p$-adic field $\mathbb{Q}_{p}$, and
$\muup_{\infty}(\lambdaup)$ to solubility over the reals. It is known that
when $d$ is sufficiently large in terms of $k$, one has

$1\lesssim\mathfrak{S}_{d,k}(\lambdaup)\lesssim 1.$ (1.3)

In particular, these bounds hold for $d\geq 4k$ when $k\geq 4$ is a power of
$2$, and for $d\geq\frac{3}{2}k$ otherwise (see Theorems 4.3 and 4.6 in
Vaughan [17]).

Throughout the paper we use the notation $f(x)\lesssim g(x)$ or $g(x)\gtrsim
f(x)$ to mean that there exists a constant $C>0$ so that $|f(x)|\leq C|g(x)|$
for all sufficiently large $x\geq 0$. The implicit constant $C$ may depend on
‘inessential’ or fixed parameters, but will be independent of ‘$x$’; below the
implicit constants will often depend on the parameters $k,d$ and $p$. For
instance, (1.3) means that there exists positive constants $C_{1}$ and $C_{2}$
depending on $d$ and $k$ so that $C_{1}\leq\mathfrak{S}_{d,k}(\lambdaup)\leq
C_{2}$.

In view of (1.2) and (1.3), we may replace the convolution
$\sigma_{\lambdaup}*f$ above by the average

$A_{\lambdaup}f(\mathbf{x}):=\lambdaup^{1-d/k}\sum_{\mathfrak{f}(\mathbf{y})=\lambdaup}f(\mathbf{x}-\mathbf{y}).$
(1.4)

Our $k$-spherical maximal function is then defined, for
$\mathbf{x}\in\mathbb{Z}^{d}$, as the pointwise supremum of all averages

$A_{*}f(\mathbf{x}):=\sup_{\lambdaup\in\mathbb{N}}|A_{\lambdaup}f(\mathbf{x})|.$
(1.5)

Variants of this maximal function were introduced by Magyar [9] and studied
later in [11, 10, 8, 2, 6, 7]. In particular, Magyar, Stein and Wainger [11]
considered the above maximal function when $k=2$ and $d\geq 5$ and proved that
it is bounded on $\ell^{p}(\mathbb{Z}^{d})$ when $p>\frac{d}{d-2}$. This
result is sharp except at the endpoint, for which the restricted weak-type
bound was proved later by Ionescu [8]. To the best of our knowledge, the
sharpest results on the boundedness of $A_{*}$ for degrees $k\geq 3$ are those
obtained by the third author in [7]. In the present paper, we give a further
improvement.

For $k\geq 3$, define

$d_{0}(k):=k^{2}-\max_{2\leq j\leq
k-1}\left\\{\frac{kj-\min(2^{j}+2,j^{2}+j)}{k-j+1}\right\\}.$ (1.6)

Further, set $\tau_{k}=\max\left\\{2^{1-k},(k^{2}-k)^{-1}\right\\}$ and define
the function $\delta_{0}(d,k)$ by

$k\,\delta_{0}(d,k):=\begin{cases}(d-d_{0})/(k^{2}+k-d_{0})&\text{if
}d_{0}(k)\leq d\leq k^{2}+k,\\\ 1+(d-k^{2}-k)\tau_{k}&\text{if
}d>k^{2}+k.\end{cases}$

Finally, define

$p_{0}(d,k):=\max\left\\{\frac{d}{d-k},1+\frac{1}{1+2\delta_{0}(d,k)}\right\\}.$

We remark that when $d>d_{0}(k)$, $p_{0}(d,k)$ always lies in the range
$(1,2)$. Our main result is the following.

######  Theorem 1.

If $k\geq 3$, $d>d_{0}(k)$ and $p>p_{0}(d,k)$, then the maximal operator
$A_{*}$, defined by (1.5), is bounded on $\ell^{p}(\mathbb{Z}^{d})$: that is,

$\|A_{*}f\|_{\ell^{p}(\mathbb{Z}^{d})}\lesssim\|f\|_{\ell^{p}(\mathbb{Z}^{d})}.$

Let $d_{0}^{*}(k):=1+\lfloor d_{0}(k)\rfloor$ denote the least dimension in
which Theorem 1 establishes that $A_{*}$ is bounded on
$\ell^{2}(\mathbb{Z}^{d})$. We emphasize that for large $k$, we have
$d_{0}^{*}(k)=k^{2}-k+O(k^{1/2})$, whereas in previous results, such as the
work of the third author [6, 7], one required $d>k^{3}-k^{2}$. While our
results improve on these previous results by a factor of the degree $k$, the
conjecture is that the maximal function is bounded on
$\ell^{2}(\mathbb{Z}^{d})$ for $d\gtrsim k$ (see [6] for details), but such a
result appears to be way beyond the reach of present methods.

It is also instructive to compare $d_{0}^{*}(k)$ to the known bounds for the
function $\tilde{G}(k)$ in the theory of Waring’s problem (defined as the
least value of $d$ for which the asymptotic formula (1.2) holds). It
transpires that the values of $d_{0}^{*}(k)$ match the best known upper bounds
on $\tilde{G}(k)$ for all but a handful of small values of $k$, and even in
those cases, we miss the best known bound on $\tilde{G}(k)$ only by a
dimension or two. For an easier comparison, we list the numerical values of
$d_{0}^{*}(k)$, $k\leq 10$, their respective analogues in earlier work, and
the bounds on $\tilde{G}(k)$ in Table 1.

$k$ | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---
$d_{0}^{*}(k)$ | 10 | 16 | 24 | 35 | 47 | 62 | 79 | 97
$d_{1}^{*}(k)$ | 13 | 33 | 81 | 181 | 295 | 449 | 649 | 901
$\tilde{G}(k)\leq$ | 8 | 15 | 23 | 34 | 47 | 61 | 78 | 97

Table 1: Comparison between the values of $d_{0}^{*}(k)$ in Theorem 1, the
corresponding values $d_{1}^{*}(k)$ in the work of Hughes [6], and the known
upper bounds on $\tilde{G}(k)$ in Bourgain [3] and Vaughan [16].

A key ingredient in the proof of Theorem 1 and its predecessors is an
approximation formula generalizing (1.2). First introduced in [11] when $k=2$,
such approximations are obtained for the average’s corresponding Fourier
multiplier:

$\widehat{A_{\lambdaup}}({\boldsymbol{\xiup}})=\lambdaup^{1-d/k}\sum_{\mathfrak{f}(\mathbf{x})=\lambdaup}e(\mathbf{x}\cdot\boldsymbol{\xiup}),$
(1.7)

where $\boldsymbol{\xiup}\in\mathbb{T}^{d}$ and $e(z)=e^{2\pi iz}$.

We need to introduce some notation in order to state our approximation
formula. Given an integer $q\geq 1$, we write
$\mathbb{Z}_{q}=\mathbb{Z}/q\mathbb{Z}$ and $\mathbb{Z}_{q}^{*}$ for the group
of units; we also write $e_{q}(x)=e(x/q)$. The $d$-dimensional Gauss sum of
degree $k$ is defined as

$G(q;a,\mathbf{b})=q^{-d}\sum_{\mathbf{x}\in\mathbb{Z}_{q}^{d}}e_{q}(a\mathfrak{f}(\mathbf{x})+\mathbf{b}\cdot\mathbf{x})$

for $a\in\mathbb{Z}_{q}$ and $\mathbf{b}\in\mathbb{Z}_{q}^{d}$. If
$\Sigma_{\lambdaup}$ denotes the surface in $\mathbb{R}_{+}^{d}$ defined by
(1.1) and $dS_{\lambdaup}(\mathbf{x})$ denotes the induced Lebesgue measure on
$\Sigma_{\lambdaup}$, we define a continuous surface measure
$d\sigma_{\lambdaup}(\mathbf{x})$ on $\Sigma_{\lambdaup}$ by

$d\sigma_{\lambdaup}(\mathbf{x}):=\lambdaup^{1-d/k}\frac{dS_{\lambdaup}(\mathbf{x})}{|\nabla\mathfrak{f}(\mathbf{x})|}.$

We note that $d\sigma_{\lambdaup}$ is essentially a probability measure on
$\Sigma_{\lambdaup}$ for all $\lambdaup$. We also fix a smooth bump function
$\psiup$, which is $1$ on $\left[-\frac{1}{8},\frac{1}{8}\right]^{d}$ and
supported in $\left[-\frac{1}{4},\frac{1}{4}\right]^{d}$. Finally, we write
$\widetilde{\muup}$ for the $\mathbb{R}^{d}$-Fourier transform of a measure
$\muup$ on $\mathbb{R}^{d}$ and $\widehat{f}$ for the $\mathbb{Z}^{d}$-Fourier
transform (which has domain $\mathbb{T}^{d}$) of a function $f$ on
$\mathbb{Z}^{d}$.

######  Theorem 2 (Approximation Formula).

If $k\geq 3$, $d>d_{0}(k)$ and $\lambdaup\in\mathbb{N}$ is sufficiently large,
then one has

$\widehat{A_{\lambdaup}}(\boldsymbol{\xiup})=\sum_{q=1}^{\infty}\sum_{a\in\mathbb{Z}_{q}^{*}}e_{q}(-\lambdaup
a)\sum_{\boldsymbol{w}\in\\{\pm
1\\}^{d}}\sum_{\mathbf{b}\in\mathbb{Z}^{d}}G(q;a,\boldsymbol{w}\mathbf{b})\psiup(q\boldsymbol{\xiup}-\mathbf{b})\widetilde{d\sigma_{\lambdaup}}(\boldsymbol{w}(\boldsymbol{\xiup}-q^{-1}\mathbf{b}))+\widehat{E_{\lambdaup}}(\boldsymbol{\xiup}),$

where the error terms $\widehat{E_{\lambdaup}}$ are the multipliers of
convolution operators satisfying the dyadic maximal inequality

$\left\|\sup_{\lambdaup\in[\Lambda/2,\Lambda)}|E_{\lambdaup}|\right\|_{\ell^{2}(\mathbb{Z}^{d})\to\ell^{2}(\mathbb{Z}^{d})}\lesssim\Lambda^{-\delta}$
(1.8)

for each $\Lambda\geq 1$ and all sufficiently small $\delta>0$.

Our Approximation Formula takes the same shape as those in [6] and [7], but
with an improved error term that relies on two recent developments:

  * •

the underlying analytic methods were improved in the authors’ previous work
[1],

  * •

and the recent resolution of the main conjecture about Vinogradov’s mean value
integral [19, 4] and related refinements of classical mean value estimates
[18, 3].

The most dramatic improvement follows from our improved analytic methods
originating in [1] where we improve the range of $\ell^{2}(\mathbb{Z}^{d})$ by
a factor of the degree $k$. In [7], this sort of improvement - which also used
the recent resolution of the Vinogradov mean values theorems [18, 3] \- was
limited to maximal functions over sufficiently sparse sequences. Here, our
bounds supersede those for integral $k$-spherical maximal functions over
sparse sequences in [7] because our treatment of the minor arcs in the error
term is more efficient. The reader may compare Lemmas 3.2 and 3.1 below to
Lemmas 2.1 and 2.2 of [7] to determine the efficacy of our method here.
Consequently, [18] and [3] allow us to further improve slightly upon a more
direct application of the Vinogradov mean value theorems from [19] and [4].
One minor drawback is that in our method the $\epsilon$-losses in [19, 4] do
not allow us to deduce endpoint bounds.

As an application, we deduce that the maximal function of the ”ergodic
Waring–Goldbach problem” introduced in our recent work [1] is bounded on the
same range of spaces as above. That maximal function is associated with
averages where, instead of sampling over integer points, we sample over points
where all coordinates are prime. To be precise, let $R^{*}(\lambdaup)$ denote
the number of prime solutions to the equation (1.1) weighted by logarithmic
factors: that is,

$R^{*}(\lambdaup):=\sum_{\mathfrak{f}(\mathbf{x})=\lambdaup}{\mathbf{1}}_{\mathbb{P}^{d}}(\mathbf{x})(\log
x_{1})\cdots(\log x_{d}),$

where $\mathbf{1}_{\mathbb{P}^{d}}$ is the indicator function of vectors
$\mathbf{x}\in\mathbb{Z}^{d}$ with all coordinates prime. When
$R^{*}(\lambdaup)>0$, define the normalized arithmetic surface measure

$\omega_{\lambdaup}({\mathbf{x}}):=\frac{1}{R^{*}(\lambdaup)}{\bf
1}_{\\{{\mathbf{y}}\in\mathbb{P}^{d}:\mathfrak{f}(\mathbf{y})=\lambdaup\\}}({\mathbf{x}})(\log
x_{1})\cdots(\log x_{d})$

and the respective convolution operators

$W_{\lambdaup}f:=\omega_{\lambdaup}*f.$ (1.9)

Similarly to (1.2), we know that as $\lambdaup\to\infty$, one has the
asymptotic

$R^{*}(\lambdaup)=\big{(}\mathfrak{S}_{d,k}^{*}(\lambdaup)+o(1)\big{)}\lambdaup^{{d/k-1}},$

where $\mathfrak{S}_{d,k}^{*}(\lambdaup)$ is a product of local densities
similar to $\mathfrak{S}_{d,k}(\lambdaup)$ above. Moreover, when $d>3k$ and
$\lambdaup$ is restricted to a particular arithmetic progression
$\Gamma_{d,k}$, we have $1\lesssim\mathfrak{S}_{d,k}^{*}(\lambdaup)\lesssim
1$, and the above estimate turns into a true asymptotic formula for
$R^{*}(\lambdaup)$ (see the introduction and references in [1]). By Theorem 6
of [1] and Theorem 1 above, we immediately obtain the following result. Here,
as in [1], $d_{1}(3)=13$ and $d_{1}(k)=k^{2}+k+3$.

######  Theorem 3.

If $k\geq 3$, $d\geq d_{1}(k)$ and $p>p_{0}(d,k)$, then the maximal function
defined by

$W_{*}f(\mathbf{x}):=\sup_{\lambdaup\in\Gamma_{d,k}}|W_{\lambdaup}f(\mathbf{x})|,$

is bounded on $\ell^{p}(\mathbb{Z}^{d})$.

As another application one may give analogous improvements of the ergodic
theorems obtained in [6], but we do not consider this here.

To establish our theorems, we follow the paradigm in [11] and strengthen the
connection to Waring’s problem as initiated in [7] by using a lemma from [1];
we then use recent work on Waring’s problem to obtain improved bounds. We
remark that [15] and [12] previously connected mean values (Hypothesis $K^{*}$
and Vinogradov’s mean value theorems respectively) to discrete fractional
integration. In Section 2, we outline the proofs of Theorems 1 and 2; we
recall some results from [6, 9] and state the key propositions required in the
proofs. The remaining sections establish the propositions. In Section 3 we
deal with the minor arcs; particularly, in Section 3.2, we use the recent work
of Bourgain, Demeter and Guth [4] on Vinogradov’s mean value theorem and a
method of Wooley [18] for estimation of mean values over minor arcs. In
Section 4, we establish the relevant major arc approximations. Finally, in
Section 5, we establish the boundedness of the maximal function associated
with the main term in the Approximation Formula.

##  2 Outline of the Proof

Since $A_{*}$ is trivially bounded on $\ell^{\infty}(\mathbb{Z}^{d})$, we may
assume through the rest of the paper that $p\leq 2$. Fix
$\Lambda\in\mathbb{N}$ and consider the dyadic maximal operator

$A_{\Lambda}f:=\sup_{\lambdaup\in[\Lambda/2,\Lambda)}|A_{\lambdaup}f|.$

When $\lambdaup\leq\Lambda$, we have

$A_{\lambdaup}f=\lambdaup^{1-d/k}\int_{\mathbb{T}}(h_{\Lambda}(\thetaup)*f)e(-\lambdaup\thetaup)\,d\thetaup,$

where

$h_{\Lambda}(\thetaup)=h_{\Lambda}(\thetaup;\mathbf{x}):=e(\thetaup\mathfrak{f}(\mathbf{x}))\mathbf{1}_{[-N,N]^{d}}(\mathbf{x})\quad\text{with}\quad
N=\Lambda^{1/k}.$

This representation allows us to decompose $A_{\lambdaup}$ into operators of
the form

$A_{\lambdaup}^{\mathfrak{B}}f=\lambdaup^{1-d/k}\int_{\mathfrak{B}}(h_{\Lambda}(\thetaup)*f)e(-\lambdaup\thetaup)\,d\thetaup,$
(2.1)

for various measurable sets $\mathfrak{B}\subseteq\mathbb{T}$.

Our decomposition of $A_{\lambdaup}$, is inspired by the Hardy–Littlewood
circle method. When $q\in\mathbb{N}$ and $0\leq a\leq q$, we define the major
arc ${\mathfrak{M}}(a/q)$ by

${\mathfrak{M}}(a/q)=\left[\frac{a}{q}-\frac{1}{4kqN^{k-1}},\frac{a}{q}+\frac{1}{4kqN^{k-1}}\right].$

We then decompose $\mathbb{T}$ into sets of _major_ and _minor arcs_ , given
by

${\mathfrak{M}}={\mathfrak{M}}(\Lambda)=\bigcup_{q\leq
N}\bigcup_{a\in\mathbb{Z}_{q}^{*}}{\mathfrak{M}}(a/q)\quad\text{ and
}\quad{\mathfrak{m}}(\Lambda)=\mathbb{T}\setminus{\mathfrak{M}}(\Lambda).$

Since the major arcs ${\mathfrak{M}}(a/q)$ are disjoint, this yields a
respective decomposition of $A_{\lambdaup}$ as

$A_{\lambdaup}=\sum_{q\leq
N}\sum_{a\in\mathbb{Z}_{q}^{*}}A_{\lambdaup}^{a/q}+A_{\lambdaup}^{{\mathfrak{m}}},$

where $A_{\lambdaup}^{a/q}:=A_{\lambdaup}^{{\mathfrak{M}}(a/q)}$.

We will use the notations $A_{*}^{\mathfrak{B}}$ and
$A_{\Lambda}^{\mathfrak{B}}$ to denote the respective maximal functions
obtained from the operators $A_{\lambdaup}^{\mathfrak{B}}$. For example, from
(2.1) and the trivial bound for the trigonometric polynomial
$h_{\Lambda}(\thetaup)*f$, we obtain the trivial $\ell^{1}$-bound

$\left\|A_{\Lambda}^{\mathfrak{B}}\right\|_{\ell^{1}(\mathbb{Z}^{d})\to\ell^{1}(\mathbb{Z}^{d})}\lesssim\Lambda|\mathfrak{B}|.$
(2.2)

In Section 3, we analyze the minor arc term and prove the following result.

######  Proposition 2.1.

If $k\geq 3$, $d>d_{0}(k)$, and $\Lambda\gtrsim 1$, then

$\left\|A_{\Lambda}^{{\mathfrak{m}}}\right\|_{\ell^{2}(\mathbb{Z}^{d})\to\ell^{2}(\mathbb{Z}^{d})}\lesssim\Lambda^{-\delta}$
(2.3)

for all $\delta\in(0,\delta_{0}(d,k))$.

For $1<p\leq 2$, interpolation between (2.2) and (2.3) yields

$\left\|A_{\Lambda}^{\mathfrak{m}}\right\|_{\ell^{p}(\mathbb{Z}^{d})\to\ell^{p}(\mathbb{Z}^{d})}\lesssim\Lambda^{-\alphaup_{p}}$

with $\alphaup_{p}=2(1+\delta)(1-1/p)-1$. When $p_{0}(d,k)<p\leq 2$, we have
$\alphaup_{p}>0$, and hence,

$\left\|A_{*}^{\mathfrak{m}}\right\|_{\ell^{p}(\mathbb{Z}^{d})\to\ell^{p}(\mathbb{Z}^{d})}\leq\sum_{\Lambda=2^{j}\gtrsim
1}\left\|A_{\Lambda}^{\mathfrak{m}}\right\|_{\ell^{p}(\mathbb{Z}^{d})\to\ell^{p}(\mathbb{Z}^{d})}\lesssim\sum_{\Lambda=2^{j}\gtrsim
1}\Lambda^{-\alphaup_{p}}\lesssim 1.$ (2.4)

The estimation of the major arc terms is more challenging, because an analogue
of (2.3) does not hold for $A_{\Lambda}^{{\mathfrak{M}}}$. Still, it is
possible to establish a slightly weaker version of Proposition 2.1. The
following result was first established by Magyar [9], for $d\geq 2^{k}$, and
then extended by the third author [6] in the present form.

######  Proposition 2.2.

If $k\geq 3$, $d>2k$, $\frac{d}{d-k}<p\leq 2$, and $\Lambda\gtrsim 1$, one has

$\left\|A_{\Lambda}^{{\mathfrak{M}}}\right\|_{\ell^{p}(\mathbb{Z}^{d})\to\ell^{p}(\mathbb{Z}^{d})}\lesssim
1.$ (2.5)

This proposition suffices to establish the $\ell^{p}$-boundedness of the
dyadic maximal functions $A_{\Lambda}$ (this is the main result of Magyar
[9]), but falls just short of what is needed for an equally quick proof of
Theorem 1. Instead, we use Theorem 2 to approximate $A_{*}^{{\mathfrak{M}}}$
by a bounded operator. Let $M_{\lambdaup}^{a/q}$ denote the convolution
operator on $\ell^{p}(\mathbb{Z}^{d})$ with Fourier multiplier

$\widehat{M_{\lambdaup}^{a/q}}(\boldsymbol{\xiup}):=e_{q}(-\lambdaup
a)\sum_{\boldsymbol{w}\in\\{\pm
1\\}^{d}}\sum_{\mathbf{b}\in\mathbb{Z}^{d}}G(q;a,\boldsymbol{w}\mathbf{b})\psiup(q\boldsymbol{\xiup}-\mathbf{b})\widetilde{d\sigma_{{\lambdaup}}}(\boldsymbol{w}(\boldsymbol{\xiup}-q^{-1}\mathbf{b})),$

and define

$M_{\lambdaup}:=\sum_{q\in\mathbb{N}}\sum_{a\in\mathbb{Z}_{q}^{*}}M_{\lambdaup}^{a/q}\quad\text{
and }\quad M_{*}:=\sup_{\lambdaup\in\mathbb{N}}|M_{\lambdaup}|.$

In Section 4, we will handle the major arc approximations and prove the
following proposition.

######  Proposition 2.3.

If $k\geq 3$, $d>2k$, and $\frac{d}{d-k}<p\leq 2$, then there exists an
exponent $\betaup_{p}=\betaup_{p}(d,k)>0$ such that

$\left\|\sup_{\lambdaup\in[\Lambda/2,\Lambda)}\big{|}A_{\lambdaup}^{{\mathfrak{M}}}-M_{\lambdaup}\big{|}\right\|_{\ell^{p}(\mathbb{Z}^{d})\to\ell^{p}(\mathbb{Z}^{d})}\lesssim\Lambda^{-\betaup_{p}}.$
(2.6)

######  Remark 2.1.

Theorem 2 is an immediate consequence of Propositions 2.1 and 2.3. Moreover,
since when $p=2$ and $d\geq\frac{5}{2}k$, inequality (2.6) holds for any
$\betaup_{2}<1/(2k)$ (see inequality (4.8) and the comments after it), the
error bound (1.8) holds for all $\delta$ with
$0<\delta<\min\\{\delta_{0}(d,k),1/(2k)\\}$.

When we sum (2.6) over dyadic $\Lambda=2^{j}$, we deduce that

$\left\|\sup_{\lambdaup\gtrsim
1}\big{|}A_{\lambdaup}^{{\mathfrak{M}}}-M_{\lambdaup}\big{|}\right\|_{\ell^{p}(\mathbb{Z}^{d})\to\ell^{p}(\mathbb{Z}^{d})}\lesssim
1.$

Combining this bound and (2.4), we conclude that the $\ell^{p}$-boundedness of
the maximal operator $A_{*}$ follows from the $\ell^{p}$-boundedness of
$M_{*}$. The following proposition, which we establish in Section 5, then
completes the proof of Theorem 1.

######  Proposition 2.4.

If $k\geq 3$, $d>2k$, and $\frac{d}{d-k}<p\leq 2$, then $M_{*}$ is bounded on
$\ell^{p}(\mathbb{Z}^{d})$.

##  3 Minor Arc Analysis

Our minor arc analysis splits naturally in two steps. The first step is a
reduction to mean value estimates related to Waring’s problem; for this we use
a technique introduced in [1]. We then apply recent work on Waring’s problem
to estimate the relevant mean values and to prove Proposition 2.1.

###  3.1 Reduction to mean value theorems

The reduction step is based on the following lemma, a special case of Lemma 7
in [1]. In the present form, the result is a slight variation of Lemma 4.2 in
[6] and is implicit also in [11].

######  Lemma 3.1.

For $\lambdaup\in\mathbb{N}$, let $T_{\lambdaup}$ be a convolution operator on
$\ell^{2}(\mathbb{Z}^{d})$ with Fourier multiplier given by

$\widehat{T_{\lambdaup}}(\boldsymbol{\xiup}):=\int_{\mathfrak{B}}K(\thetaup;\boldsymbol{\xiup})e(-\lambdaup\thetaup)\,d\thetaup,$

where $\mathfrak{B}\subseteq\mathbb{T}$ is a measurable set and
$K(\cdot;\boldsymbol{\xiup})\in L^{1}(\mathbb{T})$ is a kernel independent of
$\lambdaup$. Further, for $\Lambda\geq 2$, define the dyadic maximal functions

$T_{*}f(\mathbf{x})=T_{*}(\mathbf{x};\Lambda):=\sup_{\lambdaup\in[\Lambda/2,\Lambda)}|T_{\lambdaup}f(\mathbf{x})|.$

Then

$\|T_{*}\|_{\ell^{2}(\mathbb{Z}^{d})\to\ell^{2}(\mathbb{Z}^{d})}\leq\int_{\mathfrak{B}}\sup_{\boldsymbol{\xiup}\in\mathbb{T}^{d}}|K(\thetaup;\boldsymbol{\xiup})|\,d\thetaup.$
(3.1)

For a measurable set $\mathfrak{B}\subset\mathbb{T}$, we have

$\widehat{A_{\lambdaup}^{\mathfrak{B}}}(\boldsymbol{\xiup})=\lambdaup^{1-d/k}\int_{\mathfrak{B}}\mathcal{F}_{N}(\thetaup;\boldsymbol{\xiup})e(-\lambdaup\thetaup)\,d\thetaup,$

where

$\mathcal{F}_{N}(\thetaup;\boldsymbol{\xiup}):=\prod_{j=1}^{d}S_{N}(\thetaup,\xiup_{j})\quad\text{with}\quad
S_{N}(\thetaup,\xiup):=\sum_{|n|\leq N}e(\thetaup|n|^{k}+\xiup n).$

Thus, in the proof of Proposition 2.1, we apply (3.1) with $K=\mathcal{F}_{N}$
and $\mathfrak{B}={\mathfrak{m}}$. The supremum over $\boldsymbol{\xiup}$ on
the right side of (3.1) then stands in the way of a direct application of
known results from analytic number theory. Our next lemma overcomes this
obstacle; its proof is a variant of the argument leading to (12) in Wooley
[18].

######  Lemma 3.2.

If $k\geq 2$, $l\leq k-1$ and $s$ are natural numbers and
$\mathfrak{B}\subseteq\mathbb{T}$ a measurable set, then

$\int_{\mathfrak{B}}\sup_{\xiup\in\mathbb{T}}|S_{N}(\thetaup,\xiup)|^{2s}\,d\thetaup\lesssim
N^{l(l+1)/2}\int_{\mathfrak{B}}\int_{\mathbb{T}^{l}}\left|\sum_{n=1}^{N}e(\thetaup
n^{k}+\xiup_{l}n^{l}+\dots+\xiup_{1}n)\right|^{2s}d\boldsymbol{\xiup}d\thetaup+1.$
(3.2)

###### Proof.

Define

$\widetilde{S}_{N}(\thetaup,\xiup)=\sum_{1\leq n\leq N}e(\thetaup n^{k}+\xiup
n).$

Since

$\sup_{\xiup\in\mathbb{T}}|S_{N}(\thetaup,\xiup)|\leq
2\sup_{\xiup\in\mathbb{T}}|\widetilde{S}_{N}(\thetaup,\xiup)|+1,$ (3.3)

it suffices to establish (3.2) with $\widetilde{S}_{N}$ in place of $S_{N}$.

Set $H_{j}=sN^{j}$. For $\mathbf{h}=(h_{1},\dots,h_{l})\in\mathbb{Z}^{l}$, we
define

$a_{\mathbf{h}}(\thetaup)=\sum_{1\leq n_{1},\dots,n_{s}\leq
N}\delta(\mathbf{n};\mathbf{h})e(\thetaup\mathfrak{f}_{s,k}(\mathbf{n})),$

where

$\delta(\mathbf{n};\mathbf{h})=\begin{cases}1&\text{if
}\mathfrak{f}_{s,j}(\mathbf{n})=h_{j}\text{ for all }j=1,\dots,l,\\\
0&\text{otherwise}.\end{cases}$

We have

$\widetilde{S}_{N}(\thetaup,\xiup)^{s}=\sum_{h_{1}\leq
H_{1}}\cdots\sum_{h_{l}\leq H_{l}}a_{\mathbf{h}}(\thetaup)e(\xiup h_{1}),$

so by applying the Cauchy–Schwarz inequality we deduce that

$\sup_{\xiup}|\widetilde{S}_{N}(\thetaup,\xiup)|^{2s}\leq H_{1}\cdots
H_{l}\sum_{h_{1}\leq H_{1}}\cdots\sum_{h_{l}\leq
H_{l}}|a_{\mathbf{h}}(\thetaup)|^{2}.$

Hence,

$\int_{\mathfrak{B}}\sup_{\xiup}|\widetilde{S}_{N}(\thetaup,\xiup)|^{2s}\,d\thetaup\lesssim
N^{l(l+1)/2}\int_{\mathfrak{B}}\sum_{h_{1}\leq H_{1}}\cdots\sum_{h_{l}\leq
H_{l}}a_{\mathbf{h}}(\thetaup)\overline{a_{\mathbf{h}}(\thetaup)}\,d\thetaup.$
(3.4)

We have

$\sum_{h_{1}\leq H_{1}}\cdots\sum_{h_{l}\leq
H_{l}}a_{\mathbf{h}}(\thetaup)\overline{a_{\mathbf{h}}(\thetaup)}=\sum_{1\leq\mathbf{n},\mathbf{m}\leq
N}e\big{(}\thetaup(\mathfrak{f}_{s,k}(\mathbf{n})-\mathfrak{f}_{s,k}(\mathbf{m}))\big{)}\sum_{h_{1}\leq
H_{1}}\cdots\sum_{h_{l}\leq
H_{l}}\delta(\mathbf{n};\mathbf{h})\delta(\mathbf{m};\mathbf{h}).$

By orthogonality,

$\displaystyle\sum_{h_{1}\leq H_{1}}\cdots\sum_{h_{l}\leq
H_{l}}\delta(\mathbf{n};\mathbf{h})\delta(\mathbf{m};\mathbf{h})$
$\displaystyle=\sum_{h_{1}\leq H_{1}}\cdots\sum_{h_{l}\leq
H_{l}}\delta(\mathbf{m};\mathbf{h})\int_{\mathbb{T}^{l}}e\bigg{(}\sum_{j=1}^{l}\xiup_{j}(\mathfrak{f}_{s,j}(\mathbf{n})-h_{j})\bigg{)}\,d\boldsymbol{\xiup}$
$\displaystyle=\int_{\mathbb{T}^{l}}e\bigg{(}\sum_{j=1}^{l}\xiup_{j}(\mathfrak{f}_{s,j}(\mathbf{n})-\mathfrak{f}_{s,j}(\mathbf{m}))\bigg{)}\,d\boldsymbol{\xiup},$

since for a fixed $\mathbf{m}$, the sum over $\mathbf{h}$ has exactly one term
(in which $h_{j}=\mathfrak{f}_{s,j}(\mathbf{m})$). Hence,

$\sum_{h_{1}\leq H_{1}}\cdots\sum_{h_{l}\leq
H_{l}}a_{\mathbf{h}}(\thetaup)\overline{a_{\mathbf{h}}(\thetaup)}=\int_{\mathbb{T}^{l}}\left|\sum_{n=1}^{N}e(\thetaup
n^{k}+\xiup_{l}n^{l}+\dots+\xiup_{1}n)\right|^{2s}d\boldsymbol{\xiup}.$ (3.5)

The lemma follows from (3.3)–(3.5). ∎

######  Remark 3.1.

With slight modifications, the argument of Lemma 3.2 yields also the following
estimate

$\int_{\mathfrak{B}}\int_{\mathbb{T}}|S_{N}(\thetaup,\xiup)|^{2s}\,d\xiup
d\thetaup\lesssim
N^{l(l+1)/2-1}\int_{\mathfrak{B}}\int_{\mathbb{T}^{l}}\left|\sum_{n=1}^{N}e(\thetaup
n^{k}+\xiup_{l}n^{l}+\dots+\xiup_{1}n)\right|^{2s}d\boldsymbol{\xiup}d\thetaup+1.$

###  3.2 Mean value theorems

We now recall several mean value estimates from the literature on Waring’s
problem. The first is implicit in the proof of Theorem 10 of Bourgain [3],
which is a variant of a well-known lemma of Hua (see Lemma 2.5 in [17]). The
present result follows from eqn. (6.6) in [3]. We note that when $l=k$, the
left side of (3.6) turns into Vinogradov’s integral $J_{s,k}(N)$ and the lemma
turns into the main result of Bourgain, Demeter and Guth [4].

######  Lemma 3.3.

If $k\geq 3$, $2\leq l\leq k$ and $s\geq\frac{1}{2}l(l+1)$ are natural
numbers, then

$\int_{\mathbb{T}}\int_{\mathbb{T}^{l-1}}\left|\sum_{n=1}^{N}e(\thetaup
n^{k}+\xiup_{l-1}n^{l-1}+\dots+\xiup_{1}n)\right|^{2s}d\boldsymbol{\xiup}d\thetaup\lesssim
N^{2s-l(l+1)/{2}+\varepsilonup}.$ (3.6)

For small $k$, we will use another variant of Hua’s lemma due to Brüdern and
Robert [5]. The following is a weak form of Lemma 5 in [5].

######  Lemma 3.4.

If $k\geq 3$ and $2\leq l\leq k$ are natural numbers, then

$\int_{\mathbb{T}^{2}}\left|\sum_{n=1}^{N}e(\thetaup n^{k}+\xiup
n)\right|^{2^{l}+2}d\xiup d\thetaup\lesssim N^{2^{l}-l+1+\varepsilonup}.$
(3.7)

Note that by Remark 3.1 (with $l-1$ in place of $l$) and Lemma 3.3 we obtain a
version of (3.7) with $l(l+1)$ in place of $2^{l}+2$. Together with Lemma 3.4,
this observation yields the following bound.

######  Corollary 3.1.

If $k\geq 3$ and $2\leq l\leq k$ are natural numbers and
$r\geq\min\\{l(l+1),2^{l}+2\\}$, then

$\int_{\mathbb{T}^{2}}\left|\sum_{n=1}^{N}e(\thetaup n^{k}+\xiup
n)\right|^{r}d\xiup d\thetaup\lesssim N^{r-l-1+\varepsilonup}.$ (3.8)

We also use a variant of Lemma 3.3 that provides extra savings when the
integration over $\thetaup$ is restricted to a set of minor arcs. Lemma 3.5, a
slight modification of Theorem 1.3 in Wooley [18], improves on (3.6) in the
case $l=k-1$. Here, $\mathfrak{m}$ is the set of minor arcs defined at the
beginning of §2.

######  Lemma 3.5.

If $k\geq 3$ and $s\geq\frac{1}{2}k(k+1)$ are natural numbers, then

$\int_{{\mathfrak{m}}}\int_{\mathbb{T}^{k-2}}\left|\sum_{n=1}^{N}e(\thetaup
n^{k}+\xiup_{k-2}n^{k-2}+\dots+\xiup_{1}n)\right|^{2s}d{\boldsymbol{\xiup}}d\thetaup\lesssim
N^{2s-k(k-1)/2-2+\varepsilonup}.$ (3.9)

###### Proof.

The main point in the proof of Theorem 2.1 in Wooley [18] is the inequality
(see p. 1495 in [18])

$\int_{{\mathfrak{m}}}\int_{\mathbb{T}^{k-2}}\left|\sum_{n=1}^{N}e(\thetaup
n^{k}+\xiup_{k-2}n^{k-2}+\dots+\xiup_{1}n)\right|^{2s}d{\boldsymbol{\xiup}}d\thetaup\lesssim
N^{k-2}(\log N)^{2s+1}J_{s,k}(2N).$

The lemma follows from this inequality and the Bourgain–Demeter–Guth bound for
Vinogradov’s integral $J_{s,k}(2N)$ (the case $l=k$ of (3.6)). ∎

Now we interpolate between the above bounds.

######  Lemma 3.6.

If $k\geq 3$ and $2\leq l\leq k-1$ are natural numbers and $r$ is real, with

$r_{1}(k,l):=k^{2}-\frac{kl-\min(l^{2}+l,2^{l}+2)}{k-l+1}\leq r\leq k^{2}+k,$
(3.10)

then

$I_{r,k}(N):=\int_{{\mathfrak{m}}}\sup_{\xiup\in\mathbb{T}}\left|S_{N}(\thetaup,\xiup)\right|^{r}d\thetaup\lesssim
N^{r-k-\delta(r)+\varepsilonup},$ (3.11)

where $\delta(r)$ is the linear function of $r$ with values $\delta(r_{1})=0$
and $\delta(k^{2}+k)=1$.

###### Proof.

Let $r_{0}=\min\\{l^{2}+l,2^{l}+2\\}$. The hypothesis on $r$ implies that
$r_{0}<r\leq k^{2}+k$, so we can find $t\in[0,1)$ such that
$r=tr_{0}+(1-t)(k^{2}+k)$. By Lemma 3.2 with $l=1$ and Corollary 3.1,

$I_{r_{0},k}(N)\lesssim N\int_{\mathbb{T}^{2}}\left|\sum_{n=1}^{N}e(\thetaup
n^{k}+\xiup n)\right|^{r_{0}}d\xiup d\thetaup\lesssim
N^{r_{0}-l+\varepsilonup}.$ (3.12)

On the other hand, Lemma 3.2 with $l=k-2$ and Lemma 3.5 yield

$I_{k(k+1),k}(N)\lesssim
N^{(k-1)(k-2)/2}N^{k(k+1)-k(k-1)/2-2+\varepsilonup}=N^{k^{2}-1+\varepsilonup}.$
(3.13)

Using Hölder’s inequality and the above bounds, we get

$\displaystyle I_{r,k}(N)$ $\displaystyle\lesssim
I_{k(k+1),k}(N)^{1-t}I_{r_{0},k}(N)^{t}\lesssim
N^{(1-t)(k^{2}-1+\varepsilonup)}N^{t(r_{0}-l+\varepsilonup)}$
$\displaystyle\lesssim
N^{(1-t)(k^{2}+k)+tr_{0}}N^{-(1-t)(k+1)-tl+\varepsilonup}=N^{r-k-1+t(k-l+1)+\varepsilonup}.$

This inequality takes the form (3.11) with $\delta(r)=1-t(k-l+1)$. Since $t$
depends linearly on $r$ and $t=0$ when $r=k^{2}+k$, $\delta(r)$ is a linear
function of $r$ with $\delta(k^{2}+k)=1$. The value of $r_{1}(k,l)$ in (3.10)
is the unique solution of the linear equation $\delta(r)=0$. ∎

###  3.3 Proof of Proposition 2.1

By Lemma 3.1 and the arithmetic-geometric mean inequality,

$\left\|\sup_{\lambdaup\in[\Lambda/2,\Lambda)}|A_{\lambdaup}^{{\mathfrak{m}}}|\right\|_{\ell^{2}(\mathbb{Z}^{d})\to\ell^{2}(\mathbb{Z}^{d})}\leq
N^{k-d}\int_{{\mathfrak{m}}}\sup_{\boldsymbol{\xiup}\in\mathbb{T}^{d}}\left|\mathcal{F}_{N}(\thetaup;\boldsymbol{\xiup})\right|\,d\thetaup\leq
N^{k-d}I_{d,k}(N),$ (3.14)

where $I_{d,k}(N)$ is the integral defined in (3.11). Thus, the proposition
will follow, if we prove the inequality

$I_{d,k}(N)\lesssim N^{d-k-\delta+\varepsilonup}$ (3.15)

with $\delta=k\delta_{0}(d,k)$.

Let $l_{0}(k)$ denote the value of $l$ for which the maximum in the definition
of $d_{0}(k)$ is attained (recall (1.6)). When $d_{0}(k)<d\leq k^{2}+k$, we
may apply (3.11) with $r=d$ and $l=l_{0}$ to deduce (3.15) with
$\delta=\delta(d)$. We now observe that when $d\leq k^{2}+k$, we have
$\delta(d)=k\delta_{0}(d,k)$ and that the hypothesis $d>d_{0}(k)$ ensures that
$\delta(d)>0$.

When $d>k^{2}+k$, we enhance our estimates with the help of the
$L^{\infty}$-bound for $S_{N}(\thetaup,\xiup)$ on the minor arcs: by combining
a classical result of Weyl (see Lemma 2.4 in Vaughan [17]) and Theorem 5 in
Bourgain [3], we have

$\sup_{(\thetaup,\xiup)\in{\mathfrak{m}}\times\mathbb{T}}\left|S_{N}(\thetaup,\xiup)\right|\lesssim
N^{1-\tau_{k}+\varepsilonup},$

$\tau_{k}$ being the quantity that appears in the definition of
$\delta_{0}(d,k)$. Thus, when $d>k^{2}+k$, we have

$I_{d,k}(N)\lesssim
N^{(d-k^{2}-k)(1-\tau_{k})+\varepsilonup}I_{k(k+1),k}(N)\lesssim
N^{d-k-1-(d-k^{2}-k)\tau_{k}+\varepsilonup}.$

We conclude that (3.15) holds with $\delta=1+(d-k^{2}-k)\tau_{k}$. ∎

##  4 Major Arc Analysis

We will proceed through a series of successive approximations to
$A_{\lambdaup}^{a/q}$, which we will define by their Fourier multipliers. Our
approximations are based on the following major arc approximation for
exponential sums that appears as part of Theorem 3 of Brüdern and Robert [5].
In this result and throughout the section, we write

$G(q;a,b):=q^{-1}\sum_{x\in\mathbb{Z}_{q}}e_{q}({ax^{k}+bx})\quad\text{ and
}\quad v_{N}(\thetaup,\xiup):=\int_{0}^{N}e(\thetaup t^{k}+\xiup t)\,dt.$

######  Lemma 4.1.

Let $\thetaup,\xiup\in\mathbb{T}$, $q\in\mathbb{N}$, $a\in\mathbb{Z}_{q}^{*}$,
and $b\in\mathbb{Z}$, and suppose that

$\left|\thetaup-\frac{a}{q}\right|\leq\frac{1}{4kqN^{k-1}},\quad\left|\xiup-\frac{b}{q}\right|\leq\frac{1}{2q}.$

Then

$\sum_{n\leq N}e(\thetaup n^{k}+\xiup
n)=G(q;a,b)v_{N}(\thetaup-a/q,\xiup-b/q)+O\big{(}q^{1-1/k+\varepsilonup}\big{)}.$

Recall the definition of $S_{N}(\thetaup,\xiup)$ from Section 3.1. When
$a/q+\thetaup$ lies on a major arc ${\mathfrak{M}}(a/q)$ and
$\xiup=b/q+\etaup$, with $|\etaup|\leq 1/(2q)$, the above lemma yields

$S_{N}(a/q+\thetaup,\xiup)=G(q;a,b)v_{N}(\thetaup,\etaup)+G(q;a,-b)v_{N}(\thetaup,-\etaup)+O\big{(}q^{1-1/k+\varepsilonup}\big{)}.$
(4.1)

We will use this approximation in conjunction with the following well-known
bounds (see Theorems 7.1 and 7.3 in Vaughan [17]):

$G(q;a,b)\lesssim q^{-1/k+\varepsilonup},\quad v_{N}(\thetaup,\etaup)\lesssim
N(1+N|\etaup|+N^{k}|\thetaup|)^{-1/k}.$ (4.2)

We will make use also of the inequality

$v_{N}(\thetaup,\etaup)\lesssim N(1+N|\etaup|)^{-1/2},$ (4.3)

which follows from the representation

$v_{N}(\thetaup,\etaup)=\frac{1}{k}\int_{0}^{N^{k}}u^{1/k-1}e(\thetaup
u+\etaup u^{1/k})\,du$

and the second-derivative bound for oscillatory integrals (see p. 334 in
[14]).

### Proof of Proposition 2.3

The bulk of the work concerns the case $p=2$ of the proposition. Since

$\widehat{A_{\lambdaup}^{a/q}}(\boldsymbol{\xiup})=\lambdaup^{1-d/k}e_{q}(-\lambdaup
a)\int_{{\mathfrak{M}}(0/q)}\mathcal{F}_{N}(a/q+\thetaup;\boldsymbol{\xiup})e(-\lambdaup\thetaup)\,d\thetaup,$

the asymptotic (4.1) suggests that the following multiplier should provide a
good approximation to $\widehat{A_{\lambdaup}^{a/q}}(\boldsymbol{\xiup})$:

$\widehat{B_{\lambdaup}^{a/q}}(\boldsymbol{\xiup}):=\lambdaup^{1-d/k}e_{q}(-\lambdaup
a)\int_{{\mathfrak{M}}(0/q)}\mathcal{G}_{N}(\thetaup;\boldsymbol{\xiup})e(-\lambdaup\thetaup)\,d\thetaup,$

where

$\mathcal{G}_{N}(\thetaup;\boldsymbol{\xiup}):=\prod_{j=1}^{d}\big{\\{}G(q;a,b_{j})v_{N}(\thetaup,\etaup_{j})+G(q;a,-b_{j})v_{N}(\thetaup,-\etaup_{j})\big{\\}},$

with $b_{j}$ the unique integer such that $-\frac{1}{2}\leq
b_{j}-q\xiup_{j}<\frac{1}{2}$ and $\etaup_{j}=\xiup_{j}-b_{j}/q$. Let
$B_{\lambdaup}^{a/q}$ denote the operator on $\ell^{2}(\mathbb{Z}^{d})$ with
the above Fourier multiplier. To estimate the $\ell^{2}$-error of
approximation of $A_{\lambdaup}^{a/q}$ by $B_{\lambdaup}^{a/q}$, we use that
when $\thetaup\in{\mathfrak{M}}(0/q)$, (4.1) and (4.2) yield

$|\mathcal{F}_{N}(a/q+\thetaup;\boldsymbol{\xiup})-\mathcal{G}_{N}(\thetaup;\boldsymbol{\xiup})|\lesssim
q^{1-d/k+\varepsilonup}N^{d-1}(1+N^{k}|\thetaup|)^{(1-d)/k}.$

Thus, if $d>k+1$, we have

$\displaystyle\int_{{\mathfrak{M}}(0/q)}\sup_{\boldsymbol{\xiup}\in\mathbb{T}^{d}}|\mathcal{F}_{N}(a/q+\thetaup;\boldsymbol{\xiup})-\mathcal{G}_{N}(\thetaup;\boldsymbol{\xiup})|\,d\thetaup$
$\displaystyle\lesssim\int_{\mathbb{R}}\frac{q^{1-d/k+\varepsilonup}N^{d-1}\,d\thetaup}{(1+N^{k}|\thetaup|)^{(d-1)/k}}$
$\displaystyle\lesssim q^{1-d/k+\varepsilonup}N^{d-k-1}.$

Lemma 3.1 then yields

$\left\|\sup_{\lambdaup\in[\Lambda/2,\Lambda)}\big{|}A_{\lambdaup}^{a/q}-B_{\lambdaup}^{a/q}\big{|}\right\|_{\ell^{2}(\mathbb{Z}^{d})\to\ell^{2}(\mathbb{Z}^{d})}\lesssim
q^{1-d/k+\varepsilonup}N^{-1}.$ (4.4)

Next, we approximate $B_{\lambdaup}^{a/q}$ by the operator
$C_{\lambdaup}^{a/q}$ with multiplier

$\widehat{C_{\lambdaup}^{a/q}}(\boldsymbol{\xiup}):=\lambdaup^{1-d/k}e_{q}(-\lambdaup
a)\sum_{\mathbf{b}\in\mathbb{Z}^{d}}\psiup(q\boldsymbol{\xiup}-\mathbf{b})\int_{{\mathfrak{M}}(0/q)}\mathcal{G}_{N}(\thetaup;\boldsymbol{\xiup})e(-\lambdaup\thetaup)\,d\thetaup.$

By the localization of $\psiup$, the above sum has at most one term in which
$\mathbf{b}$ matches the integer vector that appears in the definition of
$\mathcal{G}_{N}(\thetaup;\boldsymbol{\xiup})$. Hence,

$\mathcal{G}_{N}(\thetaup;\boldsymbol{\xiup})\bigg{(}1-\sum_{\mathbf{b}\in\mathbb{Z}^{d}}\psiup(q\boldsymbol{\xiup}-\mathbf{b})\bigg{)}$

is supported on a set where $\frac{1}{8}\leq|q\xiup_{j}-b_{j}|\leq\frac{1}{2}$
for some $j$. For such $j$, by (4.3),

$v_{N}(\thetaup,\xiup_{j}-b_{j}/q)\lesssim(qN)^{1/2},$

and we conclude that

$|\mathcal{G}_{N}(\thetaup;\boldsymbol{\xiup})|\bigg{(}1-\sum_{\mathbf{b}\in\mathbb{Z}^{d}}\psiup(q\boldsymbol{\xiup}-\mathbf{b})\bigg{)}\lesssim
q^{1/2-d/k+\varepsilonup}N^{d-1/2}(1+N^{k}|\thetaup|)^{(1-d)/k}.$

So, when $d>k+1$, Lemma 3.1 gives

$\displaystyle\left\|\sup_{\lambdaup\in[\Lambda/2,\Lambda)}\big{|}B_{\lambdaup}^{a/q}-C_{\lambdaup}^{a/q}\big{|}\right\|_{\ell^{2}(\mathbb{Z}^{d})\to\ell^{2}(\mathbb{Z}^{d})}$
$\displaystyle\lesssim\int_{\mathbb{R}}\frac{q^{1/2-d/k+\varepsilonup}N^{k-1/2}\,d\thetaup}{(1+N^{k}|\thetaup|)^{(d-1)/k}}$
$\displaystyle\lesssim q^{1/2-d/k+\varepsilonup}N^{-1/2}.$ (4.5)

In our final approximation, we replace $C_{\lambdaup}^{a/q}$ by the operator
$D_{\lambdaup}^{a/q}$ with multiplier

$\widehat{D_{\lambdaup}^{a/q}}(\boldsymbol{\xiup}):=\lambdaup^{1-d/k}e_{q}(-\lambdaup
a)\sum_{\boldsymbol{w}\in\\{\pm
1\\}^{d}}\sum_{\mathbf{b}\in\mathbb{Z}^{d}}\psiup(q\boldsymbol{\xiup}-\mathbf{b})G(q;a,\boldsymbol{w}\mathbf{b})J_{\lambdaup}(\boldsymbol{w}(\boldsymbol{\xiup}-q^{-1}\mathbf{b})),$

where $\boldsymbol{w}\mathbf{b}=(w_{1}b_{1},\dots,w_{d}b_{d})$ and

$J_{\lambdaup}(\boldsymbol{\etaup}):=\int_{\mathbb{R}}\bigg{\\{}\prod_{j=1}^{d}v_{N}(\thetaup;\etaup_{j})\bigg{\\}}e(-\lambdaup\thetaup)\,d\thetaup.$

We remark that $\widehat{C_{\lambdaup}^{a/q}}(\boldsymbol{\xiup})$ can be
expressed in a matching form, with
$J_{\lambdaup}(\boldsymbol{\xiup}-q^{-1}\boldsymbol{w}\mathbf{b})$ replaced by
the analogous integral over ${\mathfrak{M}}(0/q)$. Thus, when $d>k$, we deduce
from Lemma 3.1 and (4.2) that

$\displaystyle\left\|\sup_{\lambdaup\in[\Lambda/2,\Lambda)}\big{|}C_{\lambdaup}^{a/q}-D_{\lambdaup}^{a/q}\big{|}\right\|_{\ell^{2}(\mathbb{Z}^{d})\to\ell^{2}(\mathbb{Z}^{d})}$
$\displaystyle\lesssim\int_{{\mathfrak{M}}(0/q)^{c}}\frac{q^{-d/k+\varepsilonup}N^{k}\,d\thetaup}{(1+N^{k}|\thetaup|)^{d/k}}$
$\displaystyle\lesssim q^{-1+\varepsilonup}N^{1-d/k}.$ (4.6)

Here, ${\mathfrak{M}}(0/q)^{c}$ denotes the complement of the interval
${\mathfrak{M}}(0/q)$ in $\mathbb{R}$.

Finally, we note that $D_{\lambdaup}^{a/q}$ is really $M_{\lambdaup}^{a/q}$.
Indeed, by the discussion on p. 498 in Stein [14] (see also Lemma 5 in Magyar
[10]), we have

$\displaystyle J_{\lambdaup}(\boldsymbol{\etaup})$
$\displaystyle=\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}\mathbf{1}_{[0,N]^{d}}(\mathbf{t})e(\boldsymbol{\etaup}\cdot\mathbf{t})e(\thetaup(\mathfrak{f}(\mathbf{t})-\lambdaup))\,d\mathbf{t}d\thetaup$
$\displaystyle=\lambdaup^{d/k-1}\int_{\mathbb{R}^{d}}\mathbf{1}_{[0,N]^{d}}(\mathbf{t})e(\boldsymbol{\etaup}\cdot\mathbf{t})\,d\sigma_{\lambdaup}(\mathbf{t})=\lambdaup^{d/k-1}\widetilde{d\sigma_{\lambdaup}}(\boldsymbol{\etaup}),$
(4.7)

since the surface measure $d\sigma_{\lambdaup}$ is supported in the cube
$[0,N]^{d}$. Combining this observation and (4.4)–(4), and summing over $a,q$,
we conclude that when $d>2k$,

$\sum_{q\leq
N}\sum_{a\in\mathbb{Z}_{q}^{*}}\left\|\sup_{\lambdaup\in[\Lambda/2,\Lambda)}\big{|}A_{\lambdaup}^{a/q}-M_{\lambdaup}^{a/q}\big{|}\right\|_{\ell^{2}(\mathbb{Z}^{d})\to\ell^{2}(\mathbb{Z}^{d})}\lesssim
N^{-\gammaup+\varepsilonup},$ (4.8)

with $\gammaup=\min\left(d/k-2,\frac{1}{2}\right)>0$. In particular, when
$d\geq\frac{5}{2}k$, we have $\gammaup=\frac{1}{2}$.

We can now finish the proof of Proposition 2.3. For brevity, we write
$p_{1}=\frac{d}{d-k}$. From (4.8), we obtain that

$\left\|\sup_{\lambdaup\in[\Lambda/2,\Lambda)}\big{|}A_{\lambdaup}^{{\mathfrak{M}}}-M_{\lambdaup}\big{|}\right\|_{\ell^{2}(\mathbb{Z}^{d})\to\ell^{2}(\mathbb{Z}^{d})}\lesssim
N^{-\gammaup+\varepsilonup}.$ (4.9)

On the other hand, we know from Propositions 2.2 and 2.4 that both
$A_{\Lambda}^{{\mathfrak{M}}}$ and $M_{*}$ are bounded on
$\ell^{p}(\mathbb{Z}^{d})$ when $p_{1}<p\leq 2$; hence,

$\left\|\sup_{\lambdaup\in[\Lambda/2,\Lambda)}\big{|}A_{\lambdaup}^{{\mathfrak{M}}}-M_{\lambdaup}\big{|}\right\|_{\ell^{p}(\mathbb{Z}^{d})\to\ell^{p}(\mathbb{Z}^{d})}\lesssim
1.$ (4.10)

When $p\in(p_{0}(d,k),2]$, we interpolate between (4.9) and (4.10) with
$p=p_{1}+\varepsilonup$ for a sufficiently small $\varepsilonup>0$. The
interpolation yields (2.6) with $\betaup_{p}>0$ that can be chosen arbitrarily
close to $2\gammaup(p-p_{1})/(kp(2-p_{1}))$. ∎

##  5 Main Term Contribution

In this section, we prove Proposition 2.4. First, we obtain
$L^{p}(\mathbb{R}^{d})$-bounds for the maximal function of the continuous
surface measures $d\sigma_{\lambdaup}$.

######  Lemma 5.1.

If $k\geq 2$, $d>\frac{3}{2}k$ and $p>\frac{2d-k}{2d-2k}$, then for all $f\in
L^{p}(\mathbb{R}^{d})$,

$\left\|\sup_{\lambdaup>0}|f*d\sigma_{\lambdaup}|\right\|_{L^{p}(\mathbb{R}^{d})}\lesssim\|f\|_{L^{p}(\mathbb{R}^{d})}.$
(5.1)

###### Proof.

We will deduce the lemma from a result of Rubio de Francia – Theorem A in [13]
– which reduces (5.1) to bounds for the Fourier transform of the measure
$d\sigma_{\lambdaup}$. First, we majorize the measure $d\sigma_{\lambdaup}$ by
a smooth one. By the choice of normalization of $d\sigma_{\lambdaup}$, we have

$f*d\sigma_{\lambdaup}(\mathbf{x})=\int_{\mathbb{R}^{d}}f(\mathbf{x}-\mathbf{y})\,d\sigma_{\lambdaup}(\mathbf{y})=\int_{\mathbb{R}^{d}}f(\mathbf{x}-t\mathbf{y})\,d\sigma_{1}(\mathbf{y}),$

where $t=\lambdaup^{1/k}$. Let $\phiup$ be a smooth function supported in
$\left[-\frac{1}{2},\frac{3}{2}\right]$ and such that
$\mathbf{1}_{[0,1]}(x)\leq\phiup(x)$, and write
$\phiup(\mathbf{x})=\phiup(x_{1})\cdots\phiup(x_{d})$. Since $d\sigma_{1}$ is
supported inside the unit cube $[0,1]^{d}$, we have

$\bigg{|}\int_{\mathbb{R}^{d}}f(\mathbf{x}-t\mathbf{y})\,d\sigma_{1}(\mathbf{y})\bigg{|}\leq\int_{\mathbb{R}^{d}}|f(\mathbf{x}-t\mathbf{y})|\phiup(\mathbf{y})\,d\sigma(\mathbf{y})=:\int_{\mathbb{R}^{d}}|f(\mathbf{x}-t\mathbf{y})|\,d\muup(\mathbf{y}),$

where $d\sigma$ is the surface measure on the smooth manifold
$x_{1}^{k}+\dots+x_{d}^{k}=1$. By Rubio de Francia’s theorem, the maximal
function

$\mathcal{A}_{t}f(\mathbf{x}):=\sup_{t>0}\int_{\mathbb{R}^{d}}|f(\mathbf{x}-t\mathbf{y})|\,d\muup(\mathbf{y})$

is bounded on $L^{p}(\mathbb{R}^{d})$, provided that

$\widetilde{\muup}(\boldsymbol{\xiup})\lesssim(|\boldsymbol{\xiup}|+1)^{-a}\quad\text{for
some }a>\min\left\\{\frac{1}{2p-2},\frac{1}{2}\right\\}.$ (5.2)

Thus, the lemma will follow, if we establish (5.2) with $a=d/k-1$ and $d>k+1$
because $a>1/2$ once $d>\frac{3}{2}k$.

We now turn to (5.2). Similarly to (4), we have

$\widetilde{\muup}(\boldsymbol{\xiup})=\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}\phiup(\mathbf{x})e(\boldsymbol{\xiup}\cdot\mathbf{x})e(\thetaup(\mathfrak{f}(\mathbf{x})-1))\,d\mathbf{x}d\thetaup=\int_{\mathbb{R}}\bigg{\\{}\prod_{j=1}^{d}v_{\phiup}(\thetaup,\xiup_{j})\bigg{\\}}e(-\thetaup)\,d\thetaup,$

where

$v_{\phiup}(\thetaup,\xiup):=\int_{\mathbb{R}}\phiup(x)e(\thetaup x^{k}+\xiup
x)\,dx.$

By the corollary on p. 334 of Stein [14], we have

$v_{\phiup}(\thetaup,\xiup)\lesssim(1+|\thetaup|)^{-1/k},$ (5.3)

uniformly in $\xiup$. On the other hand, if $k2^{k}|\thetaup|\leq|\xiup|$, we
have

$\bigg{|}\frac{d}{dx}\big{(}\xiup^{-1}\thetaup
x^{k}+x\big{)}\bigg{|}\geq\frac{1}{2},\quad\bigg{|}\frac{d^{j}}{dx^{j}}\big{(}\xiup^{-1}\thetaup
x^{k}+x\big{)}\bigg{|}\lesssim_{j}1$

on the support of $\phiup$. Hence, Proposition VIII.1 on p. 331 of Stein [14]
yields

$v_{\phiup}(\thetaup,\xiup)\lesssim_{M}(1+|\xiup|)^{-M}$ (5.4)

for any fixed $M\geq 1$.

We now choose an index $j$, $1\leq i\leq d$, such that
$|\boldsymbol{\xiup}|\leq d|\xiup_{i}|$ and set
$\thetaup_{0}=|\xiup_{i}|/(k2^{k})$. We apply (5.3) to the trigonometric
integrals $v_{\phiup}(\thetaup,\xiup_{j})$, $j\neq i$, and to
$v_{\phiup}(\thetaup,\xiup_{i})$ when $|\thetaup|>\thetaup_{0}$; we apply
(5.4) to $v_{\phiup}(\thetaup,\xiup_{i})$ when $|\thetaup|\leq\thetaup_{0}$.
From these bounds and the integral representation for
$\widetilde{\muup}(\boldsymbol{\xiup})$, we obtain

$\displaystyle\widetilde{\muup}(\boldsymbol{\xiup})$
$\displaystyle\lesssim\int_{|\thetaup|\leq\thetaup_{0}}\frac{(1+|\xiup_{i}|)^{-M}}{(1+|\thetaup|)^{(d-1)/k}}\,d\thetaup+\int_{|\thetaup|>\thetaup_{0}}\frac{d\thetaup}{(1+|\thetaup|)^{d/k}}$
$\displaystyle\lesssim(1+|\xiup_{i}|)^{-M}+(1+|\xiup_{i}|)^{1-d/k}\lesssim(1+|\boldsymbol{\xiup}|)^{1-d/k},$

provided that $M\geq d/k-1$ and $d>k+1$. This completes the proof. ∎

### Proof of Proposition 2.4

To prove Proposition 2.4, it suffices to prove that (uniformly in $a$ and $q$)

$\left\|\sup_{\lambdaup\in\mathbb{N}}\big{|}M_{\lambdaup}^{a/q}\big{|}\right\|_{\ell^{p}(\mathbb{Z}^{d})\to\ell^{p}(\mathbb{Z}^{d})}\lesssim
q^{-\frac{d}{k}(2-2/p)+\varepsilonup},$

for all $\frac{d}{d-k}<p\leq 2$ and $d>k$. The proposition then follows by
summing over $a$ and $q$ (the hypothesis on $p$ ensures that the resulting
series over $q$ is convergent).

Fix $q\in\mathbb{N}$ and $a\in\mathbb{Z}_{q}^{*}$ and write
$\psiup_{1}(\boldsymbol{\xiup})=\psiup(\boldsymbol{\xiup}/2)$ (so that
$\psiup=\psiup\psiup_{1}$). We borrow a trick from Magyar, Stein and Wainger
[11] to express $\widehat{M_{\lambdaup}^{a/q}}(\boldsymbol{\xiup})$ as a
linear combination of Fourier multipliers that separate the dependence on
$\lambdaup$ and from the dependence on $a/q$:

$\displaystyle\widehat{M_{\lambdaup}^{a/q}}(\boldsymbol{\xiup})$
$\displaystyle=e_{q}(-\lambdaup a)\sum_{\boldsymbol{w}\in\\{\pm
1\\}^{d}}\sum_{\mathbf{b}\in\mathbb{Z}^{d}}\psiup(q\boldsymbol{\xiup}-\mathbf{b})G(q;a,\boldsymbol{w}\mathbf{b})\widetilde{d\sigma_{\lambdaup}}(\boldsymbol{w}(\boldsymbol{\xiup}-q^{-1}\mathbf{b}))$
$\displaystyle=e_{q}(-\lambdaup a)\sum_{\boldsymbol{w}\in\\{\pm
1\\}^{d}}\bigg{(}\sum_{\mathbf{b}\in\mathbb{Z}^{d}}\psiup_{1}(q\boldsymbol{\xiup}-\mathbf{b})G(q;a,\boldsymbol{w}\mathbf{b})\bigg{)}\bigg{(}\sum_{\mathbf{b}\in\mathbb{Z}^{d}}\psiup(q\boldsymbol{\xiup}-\mathbf{b})\widetilde{d\sigma_{\lambdaup}}(\boldsymbol{w}(\boldsymbol{\xiup}-q^{-1}\mathbf{b}))\bigg{)}$
$\displaystyle=:e_{q}(-\lambdaup a)\sum_{\boldsymbol{w}\in\\{\pm
1\\}^{d}}\widehat{S_{\boldsymbol{w}}^{a/q}}(\boldsymbol{\xiup})\widehat{T_{\lambdaup,\boldsymbol{w}}^{q}}(\boldsymbol{\xiup}).$

Since $\boldsymbol{w}$ takes on precisely $2^{d}$ values for each $a/q$, it
suffices to prove that

$\left\|\sup_{\lambdaup\in\mathbb{N}}\big{|}T_{\lambdaup,\boldsymbol{w}}^{q}\circ
S_{\boldsymbol{w}}^{a/q}\big{|}\right\|_{\ell^{p}(\mathbb{Z}^{d})\to\ell^{p}(\mathbb{Z}^{d})}\lesssim
q^{-\frac{d}{k}(2-2/p)+\varepsilonup},$ (5.5)

uniformly for $\boldsymbol{w}\in\\{-1,1\\}^{d}$. To prove (5.5), we will first
bound the maximal function over the ‘Archimedean’ multipliers
$T_{\lambdaup,\boldsymbol{w}}^{q}$, and then we will bound the non-Archimedean
multipliers $S_{\boldsymbol{w}}^{a/q}$. This is possible because
$S_{\boldsymbol{w}}^{a/q}$ is independent of $\lambdaup\in\mathbb{N}$.

For $d>\frac{3}{2}k$ and $p>\frac{2d-k}{2d-2k}$, Lemma 5.1 and Corollary 2.1
of [11] (the ‘Magyar–Stein–Wainger transference principle’) give the bound

$\left\|\sup_{\lambdaup\in\mathbb{N}}\big{|}T_{\lambdaup,\boldsymbol{w}}^{q}g\big{|}\right\|_{\ell^{p}(\mathbb{Z}^{d})}\lesssim\|g\|_{\ell^{p}(\mathbb{Z}^{d})},$
(5.6)

with an implicit constant independent of $q$ and $\boldsymbol{w}$. We now
observe that $S_{\boldsymbol{w}}^{a/q}$ does not depend on $\lambdaup$ and
apply (5.6) with $g=S_{\boldsymbol{w}}^{a/q}f$ to find that

$\left\|\sup_{\lambdaup\in\mathbb{N}}\big{|}T_{\lambdaup,\boldsymbol{w}}^{q}\circ
S_{\boldsymbol{w}}^{a/q}\big{|}\right\|_{\ell^{p}(\mathbb{Z}^{d})\to\ell^{p}(\mathbb{Z}^{d})}\lesssim\big{\|}S_{\boldsymbol{w}}^{a/q}\big{\|}_{\ell^{p}(\mathbb{Z}^{d})\to\ell^{p}(\mathbb{Z}^{d})},$
(5.7)

under the same assumptions on $d$ and $p$ which we note are weaker than the
hypotheses of the proposition. Finally, observe that $G(q;a,b)$ is a
$q$-periodic function with $\mathbb{Z}_{q}$-Fourier transform equal to

$\sum_{b\in\mathbb{Z}_{q}}e_{q}(-mb)G(q;a,b)=q^{-1}\sum_{x\in\mathbb{Z}_{q}}e_{q}(ax^{k})\sum_{b\in\mathbb{Z}_{q}}e_{q}(b(x-m))=e_{q}({am^{k}}),$

for each $m\in\mathbb{Z}_{q}$. Hence, we may apply Proposition 2.2 in [11] and
the bound (4.2) for $G(q;a,b)$ to deduce that

$\big{\|}S_{\boldsymbol{w}}^{a/q}\big{\|}_{\ell^{p}(\mathbb{Z}^{d})\to\ell^{p}(\mathbb{Z}^{d})}\lesssim
q^{-\frac{d}{k}(2-2/p)+\varepsilonup}.$ (5.8)

The desired inequality (5.5) follows immediately from (5.7) and (5.8). ∎

## Acknowledgments

The first author was supported by NSF grant DMS-1502464. The second author was
supported by NSF grant DMS-1147523 and by the Fields Institute. Last but not
least, the authors thank Alex Nagel and Trevor Wooley for several helpful
discussions.

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[TA] Theresa C. Anderson
Department of Mathematics
University of Wisconsin–Madison
Madison, WI 53705, U.S.A.
tcandersonmathwiscedu [BC] Brian Cook
Department of Mathematicl Sciences
Kent State University
Kent, OH 44242
U.S.A.
bcook25kentedu [KH] Kevin Hughes
School of Mathematics
The University of Bristol
Bristol, BS8 1TW, UK
Kevin.Hughesbristolacuk [AK] Angel Kumchev
Department of Mathematics
Towson University
Towson, MD 21252, U.S.A.
akumchevtowsonedu