# SVTR: Scene Text Recognition with a Single Visual Model

Yongkun Du1 Zhineng Chen2111Corresponding Author Caiyan Jia1 Xiaoting Yin3
Tianlun Zheng2
Chenxia Li3 Yuning Du3 Yu-Gang Jiang2 1School of Computer and Information
Technology and Beijing Key Lab of Traffic Data Analysis and Mining, Beijing
Jiaotong University, China
2Shanghai Collaborative Innovation Center of Intelligent Visual Computing,
School of Computer Science, Fudan University, China
3Baidu Inc., China<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Dominant scene text recognition models commonly contain two building blocks, a
visual model for feature extraction and a sequence model for text
transcription. This hybrid architecture, although accurate, is complex and
less efficient. In this study, we propose a Single Visual model for Scene Text
recognition within the patch-wise image tokenization framework, which
dispenses with the sequential modeling entirely. The method, termed SVTR,
firstly decomposes an image text into small patches named character
components. Afterward, hierarchical stages are recurrently carried out by
component-level mixing, merging and/or combining. Global and local mixing
blocks are devised to perceive the inter-character and intra-character
patterns, leading to a multi-grained character component perception. Thus,
characters are recognized by a simple linear prediction. Experimental results
on both English and Chinese scene text recognition tasks demonstrate the
effectiveness of SVTR. SVTR-L (Large) achieves highly competitive accuracy in
English and outperforms existing methods by a large margin in Chinese, while
running faster. In addition, SVTR-T (Tiny) is an effective and much smaller
model, which shows appealing speed at inference. The code is publicly
available at https://github.com/PaddlePaddle/PaddleOCR.

##  1 Introduction

Scene text recognition aims to transcript a text in natural image to digital
character sequence, which conveys high-level semantics vital for scene
understanding. The task is challenging due to variations in text deformations,
fonts, occlusions, cluttered background, etc. In the past years, many efforts
have been made to improve the recognition accuracy. Modern text recognizers,
besides accuracy, also take factors like inference speed into account because
of practical requirements.

Figure 1: (a) CNN-RNN based models. (b) Encoder-Decoder models. MHSA and MHA
denote multi-head self-attention and multi-head attention, respectively. (c)
Vision-Language models. (d) Our SVTR, which recognizes scene text with a
single visual model and enjoys efficient, accurate and cross-lingual
versatile.

Methodologically, scene text recognition can be viewed as a cross-modal
mapping from image to character sequence. Typically, the recognizer consists
of two building blocks, a visual model for feature extraction and a sequence
model for text transcription. For example, CNN-RNN based models Zhai et al.
(2016); Shi et al. (2017) first employed CNN for feature extraction. The
feature was then reshaped as a sequence and modeled by BiLSTM and CTC loss to
get the prediction (Figure 1(a)). They are featured by efficiency and remain
the choice for some commercial recognizers. However, the reshaping is
sensitive to text disturbances such as deformation, occlusion, etc, limiting
their effectiveness.

Figure 2: Overall architecture of the proposed SVTR. It is a three-stage
height progressively decreased network. In each stage, a series of mixing
blocks are carried out and followed by a merging or combining operation. At
last, the recognition is conducted by a linear prediction.

Later, encoder-decoder based auto-regressive methods became popular Sheng et
al. (2019); Li et al. (2019); Zheng et al. (2021), the methods transform the
recognition as an iterative decoding procedure (Figure 1(b)). As a result,
improved accuracy was obtained as the context information was considered.
However, the inference speed is slow due to the character-by-character
transcription. The pipeline was further extended to vision-language based
framework Yu et al. (2020); Fang et al. (2021), where language knowledge was
incorporated (Figure 1(c)) and parallel prediction was conducted. However, the
pipeline often requires a large capacity model or complex recognition paradigm
to ensure the recognition accuracy, restricting its efficiency.

Recently, there are efforts emphasized developing simplified architectures to
accelerate the speed. For example, using complex training paradigm but simple
model for inference. CRNN-RNN based solution was revisited in Hu et al.
(2020). It utilized the attention mechanism and graph neural network to
aggregate sequential features corresponding to the same character. At
inference, the attention modeling branch was discarded to balance accuracy and
speed. PREN2D Yan et al. (2021) further simplified the recognition by
aggregating and decoding the 1D sub-character features simultaneously. Wang et
al. (2021) proposed VisionLAN. It introduced a character-wise occluded
learning to endue the visual model with language capability. While at
inference, the visual model was applied merely for speedup. In view of the
simplicity of a single visual model based architecture, some recognizers were
proposed by employing off-the-shelf CNN Borisyuk et al. (2018) or ViT Atienza
(2021) as the feature extractor. Despite being efficiency, their accuracy is
less competitive compared to state-of-the-art methods.

We argue that the single visual model based scheme is effective only if
discriminative character features could be extracted. Specifically, the model
could successfully catch both intra-character local patterns and inter-
character long-term dependence. The former encodes stroke-like features that
describe fine-grained features of a character, being a critical source for
distinguishing characters. While the latter records language-analogous
knowledge that describes the characters from a complementary aspect. However,
the two properties are not well modeled by previous feature extractors. For
example, CNN backbones are good at modeling local correlation rather than
global dependence. Meanwhile, current transformer-based general-purpose
backbones would not give privilege to local character patterns.

Motivated by the issues mentioned above, this work aims to enhance the
recognition capability by reinforcing the visual model. To this end, we
propose a visual-based model SVTR for accurate, fast and cross-lingual
versatile scene text recognition. Inspired by the recent success of vision
transformer Dosovitskiy et al. (2021); Liu et al. (2021), SVTR first
decomposes an image text into small 2D patches termed character components, as
each of which may contain a part of a character only. Thus, patch-wise image
tokenization followed by self-attention is applied to capture recognition
clues among character components. Specifically, a text-customized architecture
is developed for this purpose. It is a three-stage height progressively
decreased backbone with mixing, merging and/or combining operations. Local and
global mixing blocks are devised and recurrently employed at each stage,
together with the merging or combining operation, acquiring the local
component-level affinities that represent the stroke-like feature of a
character, and long-term dependence among different characters. Therefore, the
backbone extracts component features of different distance and at multiple
scales, forming a multi-grained character feature perception. As a result, the
recognition is reached by a simple linear prediction. In the whole process
only one visual model is employed. We construct four architecture variants
with varying capacities. Extensive experiments are carried out on both English
and Chinese scene text recognition tasks. It is shown that SVTR-L (large)
achieves highly competitive results in English and outperforms state-of-the-
art models by a large margin in Chinese, while running faster. Meanwhile,
SVTR-T (Tiny) is also an effective and much smaller model yet with appealing
inference speed. The main contributions of this work can be summarized as
follows.

  * •

We demonstrate, for the first time, that a single visual model can achieve
competitive or even higher accuracy as advanced vision-language models in
scene text recognition. It is promising to practical applications due to its
efficiency and cross-lingual versatility.

  * •

We propose SVTR, a text-customized model for recognition. It introduces local
and global mixing blocks for extracting stroke-like features and inter-
character dependence, respectively, together with the multi-scale backbone,
forming a multi-grained feature description.

  * •

Empirical studies on public benchmarks demonstrate the superiority of SVTR.
SVTR-L achieves the state-of-the-art performance in recognizing both English
and Chinese scene texts. While SVTR-T is effective yet efficient, with
parameters of 6.03M and consuming 4.5ms per image text on average in one
NVIDIA 1080Ti GPU.

##  2 Method

###  2.1 Overall Architecture

Overview of the proposed SVTR is illustrated in Figure 2. It is a three-stage
height progressively decreased network dedicated to text recognition. For an
image text of size $H\times W\times 3$, it is first transformed to
$\frac{H}{4}\times\frac{W}{4}$ patches of dimension $D_{0}$ via a progressive
overlapping patch embedding. The patches are termed character components, each
associating with a fraction of text character in the image. Then, three
stages, each composed of a series of mixing blocks followed by a merging or
combining operation, are carried out at different scales for feature
extraction. local and global mixing blocks are devised for stroke-like local
pattern extraction and inter-component dependence capturing. With the
backbone, component feature and dependence of different distance and at
multiple scales are characterized, generating a representation referred to as
$C$ of size $1\times\frac{W}{4}\times D_{3}$, which perceives multi-grained
character features. Finally, a parallel linear prediction with de-duplication
is conducted to get the character sequence.

Figure 3: (a) The linear projection in ViT Dosovitskiy et al. (2021). (b) Our
progressive overlapping patch embedding.

###  2.2 Progressive Overlapping Patch Embedding

With an image text, the first task is to obtain feature patches which
represent character components from $X\in R^{H\times W\times 3}$ to $CC_{0}\in
R^{\frac{H}{4}\times\frac{W}{4}\times D_{0}}$. There exists two common one-
step projections for this purpose, i.e., a $4\times 4$ disjoint linear
projection (see Figure 3(a)) and a $7\times 7$ convolution with stride 4.
Alternatively, we implement the patch embedding by using two consecutive
$3\times 3$ convolutions with stride 2 and batch normalization, as shown in
Figure 3(b). The scheme, despite increasing the computational cost a little,
adds the feature dimension progressively which is in favor of feature fusion.
Ablation study in Section 3.3 shows its effectiveness.

###  2.3 Mixing Block

Since two characters may differ slightly, text recognition heavily relies on
features at character component level. However, existing studies mostly employ
a feature sequence to represent the image text. Each feature corresponds to a
thin-slice image region, which is often noisy especially for irregular text.
It is not optimal for describing the character. Recent advancement of vision
transformer introduces 2D feature representation, but how to leverage this
representation in the context of text recognition is still worthy
investigation.

More specifically, with the embedded components, we argue that text
recognition requires two kinds of features. The first is local component
patterns such as the stroke-like feature. It encodes the morphology feature
and correlation between different parts of a character. The second is inter-
character dependence such as the correlation between different characters or
between text and non-text components. Therefore, we devise two mixing blocks
to perceive the correlation by using self-attention with different reception
fields.

##### Global Mixing.

As seen in Figure 4(a), global mixing evaluates the dependence among all
character components. Since text and non-text are two major elements in an
image, such a general-purpose mixing could establish the long-term dependence
among component from different characters. Besides, it also capable of
weakening the influence of non-text components, while enhancing the importance
of text components. Mathematically, for character components $CC_{i-1}$ from
the previous stage, it is first reshaped as a feature sequence. When feeding
into the mixing block, a layer norm is applied and followed by a multi-head
self-attention for dependence modeling. In the following, a layer norm and a
MLP are sequentially applied for feature fusion. Together with the shortcut
connections, forming the global mixing block.

Figure 4: Illustration of (a) global mixing and (b) local mixing.

##### Local Mixing.

As seen in Figure 4(b), local mixing evaluates the correlation among
components within a predefined window. Its objective is to encode the
character morphology feature and establish the associations between components
within a character, which simulates the stroke-like feature vital for
character identification. Different from global mixing, local mixing considers
a neighborhood for each component. Similar to convolution, the mixing is
carried out in a sliding window manner. The window size is empirically set to
$7\times 11$. Compared with the global mixing, it implements the self-
attention mechanism to capture local patterns.

As aforementioned, the two mixing blocks aims to extract different features
that are complementary. In SVTR, the blocks are recurrently applied many times
in each stage for comprehensive feature extraction. Permutation of the two
kinds of blocks will be ablated later.

###  2.4 Merging

It is computational expensive to maintain a constant spatial resolution across
stages, which also leads to redundant representation. As a consequence, we
devise a merging operation following the mixing blocks at each stage (except
the last one). With the features outputted from the last mixing block, we
first reshape it to an embedding of size $h\times w\times d_{i-1}$, denoting
current height, width and channels, respectively. Then, we employ a $3\times
3$ convolution with stride 2 in the height dimension and 1 in the width
dimension, followed by a layer norm, generating an embedding of size
$\frac{h}{2}\times w\times d_{i}$.

The merging operation halve the height while keep a constant width. It not
only reduce the computational cost, but also build a text-customized
hierarchical structure. Typically, most image text appears horizontally or
near horizontally. Compressing the height dimension can establish a multi-
scale representation for each character, while not affecting the patch layout
in the width dimension. Therefore, it would not increase the chance of
encoding adjacent characters into the same component across stages. We also
increase the channel dimension $d_{i}$ to compensate the information loss.

###  2.5 Combining and Prediction

In the last stage, the merging operation is replaced by a combining operation.
It pools the height dimension to 1 at first, followed by a fully-connected
layer, non-linear activation and dropout. By doing this, character components
are further compressed to a feature sequence, where each element is
represented by a feature of length $D_{3}$. Compared to the merging operation,
the combining operation can avoid applying convolution to an embedding whose
size is very small in one dimension, e.g., with 2 in height.

With the combined feature, we implement the recognition by a simple parallel
linear prediction. Concretely, a linear classifier with $N$ nodes is employed.
It generates a transcript sequence of size $\frac{W}{4}$, where ideally,
components of the same character are transcribed to duplicate characters,
components of non-text are transcribed to a blank symbol. The sequence is
automatically condensed to the final result. In the implementation, $N$ is set
to 37 for English and 6625 for Chinese.

###  2.6 Architecture Variants

There are several hyper-parameters in SVTR, including the depth of channel and
the number of heads at each stage, the number of mixing blocks and their
permutation. By varying them, SVTR architectures with different capacities
could be obtained and we construct four typical ones, i.e., SVTR-T (Tiny),
SVTR-S (Small), SVTR-B (Base) and SVTR-L (Large). Their detail configurations
are shown in Table 1.

Models | $\left[D_{0},D_{1},D_{2}\right]$ | $[L_{1},L_{2},L_{3}]$ | Heads | $D_{3}$ | Permutation | Params (M) | FLOPs (G)
---|---|---|---|---|---|---|---
SVTR-T | [64,128,256] | [3,6,3] | [2,4,8] | 192 | $[L]_{6}{[G]_{6}}$ | 4.15 | 0.29
SVTR-S | [96,192,256] | [3,6,6] | [3,6,8] | 192 | $[L]_{8}{[G]_{7}}$ | 8.45 | 0.63
SVTR-B | [128,256,384] | [3,6,9] | [4,8,12] | 256 | $[L]_{8}{[G]_{10}}$ | 22.66 | 3.55
SVTR-L | [192,256,512] | [3,9,9] | [6,8,16] | 384 | $[L]_{10}{[G]_{11}}$ | 38.81 | 6.07
Table 1: Architecture specifications of SVTR variants (w/o counting the
rectification module and linear classifier).

##  3 Experiments

###  3.1 Datasets

For English recognition task, our models are trained on two commonly used
synthetic scene text datasets, i.e., MJSynth (MJ) Jaderberg et al. (2014,
2015) and SynthText (ST) Gupta et al. (2016). Then the models are tested on
six public benchmarks as follows: ICDAR 2013 (IC13) KaratzasAU et al. (2013)
contains 1095 testing images. we discard images that contain non-alphanumeric
characters or less than three characters. As a result, IC13 contains 857
images. Street View Text (SVT) Wang et al. (2011) has 647 testing images
cropped form Google Street View. Many images are severely corrupted by noise,
blur, and low resolution. IIIT5K-Words (IIIT) Mishra et al. (2012) is
collected from the website and contains 3000 testing images. ICDAR 2015 (IC15)
Karatzas et al. (2015) is taken with Google Glasses without careful position
and focusing. IC15 has two versions with 1,811 images and 2,077 images. We use
the former one. Street View Text-Perspective (SVTP) Phan et al. (2013) is also
cropped form Google Street View. There are 639 test images in this set and
many of them are perspectively distorted. CUTE80 (CUTE) is proposed inAnhar et
al. (2014) for curved text recognition. 288 testing images are cropped from
full images by using annotated words.

For Chinese recognition task, we use the Chinese Scene Dataset Chen et al.
(2021). It is a public dataset containing 509,164, 63,645 and 63,646 training,
validation, and test images. The validation set is utilized to determine the
best model, which is then assessed using the test set.

###  3.2 Implementation Details

SVTR uses the rectification module Shi et al. (2019), where the image text is
resized to $32\times 64$ for distortion correction. We use the AdamW optimizer
with weight decay of 0.05 for training. For English models, the initial
learning rate are set to $\frac{5}{10^{4}}\times\frac{batchsize}{2048}$. The
cosine learning rate scheduler with 2 epochs linear warm-up is used in all 21
epochs. Data augmentation like rotation, perspective distortion, motion blur
and Gaussian noise, are randomly performed during the training. The alphabet
includes all case-insensitive alphanumerics. The maximum predict length is set
to 25. The length exceeds the vast majority of English words. For Chinese
models, the initial learning rate are set to
$\frac{3}{10^{4}}\times\frac{batchsize}{512}$. The cosine learning rate
scheduler with 5 epochs linear warm-up is used in all 100 epochs. Data
augmentation was not used for training. The maximum predict length is set to
40. Word accuracy is used as the evaluation metric. All models are trained by
using 4 Tesla V100 GPUs on PaddlePaddle.

###  3.3 Ablation Study

To better understand SVTR, we perform controlled experiments on both IC13
(regular text) and IC15 (irregular text) under different settings. For
efficiency, all the experiments are carried out by using SVTR-T without
rectification module and data augmentation.

####  3.3.1 The Effectiveness of Patch Embedding

As seen in Table 2 (the left half), different embedding strategies behave
slightly different in recognition accuracy. Our progressive embedding scheme
outperforms the two default ones by 0.75% and 2.8% on average on the two
datasets, indicating that it is effective especially for irregular text.

####  3.3.2 The Effectiveness of Merging

There are also two choices, i.e., applying the merging operation to build a
progressively decreased network ($prog$), and keeping a constant spatial
resolution across stages ($none$). As indicated in the right half of Table 2.
The merging not only reduces the computational cost, but also increases the
accuracy on both datasets. It verifies that a multi-scale sampling on the
height dimension is effective for text recognition.

Embedding | IC13 | IC15 | Merging | IC13 | IC15 | FLOPS
---|---|---|---|---|---|---
Linear | 92.5 | 72.0 | None | 92.4 | 71.8 | 1.10
Overlap | 93.0 | 73.9 | Prog | 93.5 | 74.8 | 0.29
Ours | 93.5 | 74.8 |  |  |  |
Table 2: Ablation study on patch embedding (the left half) and merging (the right half). CP | IC13 | IC15 | CP | IC13 | IC15
---|---|---|---|---|---
None | 91.6 | 68.2 | ${[}G{]}_{12}$ | 92.7 | 73.7
${[}G{]}_{6}{[}L{]}_{6}$ | 92.2 | 71.9 | ${[}L{]}_{12}$ | 91.3 | 70.2
${[}L{]}_{6}{[}G{]}_{6}$ | 93.5 | 74.8 | ${[}GL{]}_{6}$ | 91.9 | 72.4
|  |  | $[LG]_{6}$ | 93.5 | 73.5
Table 3: Ablation study on mixing block permutation.

####  3.3.3 The Permutation of Mixing Blocks

There are various ways to group the global and local mixing blocks in each
stage. Several of them are assessed in Table 3, where $none$ means no mixing
block is taken into account. ${[}G{]}_{6}{[}L{]}_{6}$ means for each stage,
six global mixing blocks are carried out at first, and then six local mixing
blocks. Others are defined similarly. As can be seen, almost every scheme
gives certain accuracy improvement. We believe that the improvements are
attributed to perceiving comprehensive character component features. The
relatively large gains on irregular text further explains the mixing block is
helpful to feature learning in complex scenarios. It is observed that
${[}L{]}_{6}{[}G{]}_{6}$ reports the best accuracy. It gives accuracy gains of
1.9% and 6.6% on IC13 and IC15 when compared with $none$. By placing the local
mixing block in front of the global one, it is beneficial to guide the global
mixing block to focus on long-term dependence capturing. On the contrary,
switching their permutation is likely to confuse the role of the global mixing
block, which may repetitively attend to local characteristics.

Figure 5: Accuracy-parameter (M) and Accuracy-speed (ms) plots of different models on IC15. method | English regular | English unregular | Chinese |  Params (M) |  Speed (ms)
---|---|---|---|---|---
IC13 | SVT | IIIT5k | IC15 | SVTP | CUTE | Scene
| CRNNShi et al. (2017) | 91.1 | 81.6 | 82.9 | 69.4 | 70.0 | 65.5 | 53.4 | 8.3 | 6.3
| RosettaBorisyuk et al. (2018) | 90.9 | 84.7 | 84.3 | 71.2 | 73.8 | 69.2 | - | 44.3 | 10.5
| SRN*Yu et al. (2020) | 93.2 | 88.1 | 92.3 | 77.5 | 79.4 | 84.7 | - | - | -
| PREN*Yan et al. (2021) | 94.7 | 92.0 | 92.1 | 79.2 | 83.9 | 81.3 | - | 29.1 | 40.0
| ViTSTRAtienza (2021) | 93.2 | 87.7 | 88.4 | 78.5 | 81.8 | 81.3 | - | 85.5 | 11.2
| ABINet*Fang et al. (2021) | 94.9 | 90.4 | 94.6 | 81.7 | 84.2 | 86.5 | - | 23.5 | 50.6
Lan-free | VST*Tang et al. (2022) | 95.6 | 91.9 | 95.6 | 82.3 | 87.0 | 91.8 | - | - | -
| ASTERShi et al. (2019) | - | 89.5 | 93.4 | 76.1 | 78.5 | 79.5 | 54.5 | 27.2 | -
| MORANLuo et al. (2019) | - | 88.3 | 91.2 | - | 76.1 | 77.4 | 51.8 | 28.5 | -
| NRTRSheng et al. (2019) | 94.7 | 88.3 | 86.5 | - | - | - | - | 31.7 | 160
| SARLi et al. (2019) | 91.0 | 84.5 | 91.5 | 69.2 | 76.4 | 83.5 | 62.5 | 57.5 | 120
| AutoSTRZhang et al. (2020) | - | 90.9 | 94.7 | 81.8 | 81.7 | 84.0 | - | 10.4 | 207
| SRNYu et al. (2020) | 95.5 | 91.5 | 94.8 | 82.7 | 85.1 | 87.8 | 60.1 | 54.7 | 25.4
| PREN2DYan et al. (2021) | 96.4 | 94.0 | 95.6 | 83.0 | 87.6 | 91.7 | - | - | -
| VisionLANWang et al. (2021) | 95.7 | 91.7 | 95.8 | 83.7 | 86.0 | 88.5 | - | 32.8 | 28.0
| ABINetFang et al. (2021) | 97.4 | 93.5 | 96.2 | 86 | 89.3 | 89.2 | - | 36.7 | 51.3
Lan-aware | VSTTang et al. (2022) | 96.4 | 93.8 | 96.3 | 85.4 | 88.7 | 95.1 | - | 64.0 | -
| SVTR-T (Tiny) | 96.3 | 91.6 | 94.4 | 84.1 | 85.4 | 88.2 | 67.9 | 6.03 | 4.5
| SVTR-S (Small) | 95.7 | 93.0 | 95.0 | 84.7 | 87.9 | 92.0 | 69.0 | 10.3 | 8.0
| SVTR-B (Base) | 97.1 | 91.5 | 96.0 | 85.2 | 89.9 | 91.7 | 71.4 | 24.6 | 8.5
Ours | SVTR-L (Large) | 97.2 | 91.7 | 96.3 | 86.6 | 88.4 | 95.1 | 72.1 | 40.8 | 18.0
Table 4: Results on six English and one Chinese benchmarks tested against
existing methods, where $CRNN$ and $Rosetta$ are from the reproduction of
CombBest Baek et al. (2019). $Lan$ means language and * means the language-
free version of the corresponding method. The speed is the inference time on
one NVIDIA 1080Ti GPU averaged over 3000 English image text.

###  3.4 Comparison with State-of-the-Art

We compare SVTR with previous works on six English benchmarks covering regular
and irregular text and one Chinese scene dataset in Table 4. The methods are
grouped by whether utilizing the language information, i.e., lan-free and lan-
aware. We first look into the result in English datasets. Even SVTR-T, the
smallest one of the SVTR family, achieves highly competitive accuracy among
the lan-free ones. While other SVTRs are new state-of-the-art on most
datasets. When further compared with lan-aware ones, SVTR-L gains the best
accuracy on IIIT, IC15 and CUTE among the six English benchmarks. Its overall
accuracy is on par with recent studies Fang et al. (2021); Tang et al. (2022),
which used extra language models. As a contrast, SVTR enjoys its simplicity
and runs faster. The results basically imply that a single visual model also
could perform the recognition well, as discriminative character features are
successfully extracted.

We then analyze the result on Chinese Scene Dataset. Chen et al. (2021) gives
the accuracy of six existing methods as shown in the table. Encouragingly,
SVTR performs considerably well. Compared with SAR, the best one among the
listed ones, accuracy gains ranging from 5.4% to 9.6% are observed, which are
noticeable improvements. The result is explained as SVTR comprehensively
perceives multi-grained character component features. They are well suited to
characterize Chinese words that have rich stroke patterns.

In Figure.5, we also depict the accuracy, parameter and inference speed of
different models on IC15. Owning to their simpleness, SVTRs consistently rank
top-tier in both accuracy-parameter and accuracy-speed plots, further
demonstrating its superiority compared to existing methods.

###  3.5 Visualization Analysis

We have visualized the attention maps of SVTR-T when decoding different
character components. Each map can be explained as serving a different role in
the whole recognition. Nine maps are selected for illustration, as shown in
Figure.6. The first line shows three maps of gazing into a fraction of
character ”B”, with emphasis on its left side, bottom and middle parts,
respectively. It indicates that different character regions are contributed to
the recognition. The second line exhibits three maps of gazing into different
characters, i.e., ”B”, ”L”, and ”S”. SVTR-T is also able to learn its feature
by viewing a character as a whole. While the third line exhibits three maps
simultaneously activate multiple characters. It implies that dependencies
among different characters are successfully captured. The three lines together
reveal that sub-character, character-level and cross-character clues are all
captured by the recognizer, in accordance with the claim that SVTR perceives
multi-grained character component features. It again explains the
effectiveness of SVTR.

With the results and discussion, we conclude with that SVTR-L enjoys the
merits of accurate, fast and versatile, being a highly competitive choice in
accuracy-oriented applications. While SVTR-T is an effective and much smaller
model yet quite fast. It is appealing in resource-limited scenarios.

Figure 6: Visualization of SVTR-T attention maps.

##  4 Conclusion

We have presented SVTR, a customized visual model for scene text recognition.
It extracts multi-grained character features that describe both stroke-like
local patterns and inter-component dependence of varying distance at multiple
height scales. Therefore, the recognition task can be conducted by using a
single visual model, enjoying the merits of accuracy, efficiency and cross-
lingual versatility. SVTR with different capacities are also devised to meet
diverse application needs. Experiments on both English and Chinese benchmarks
basically verify the proposed SVTR. Highly competitive or even better accuracy
is observed compared to state-of-the-art methods, while running faster. We
hope that SVTR will foster further research in scene text recognition.

## Acknowledgments

The work is supported by the National Nature Foundation of China (No.
62172103, 61876016) and CCF-Baidu Open Fund (No.2021PP15002000).

## References

  * Anhar et al. [2014] R. Anhar, S. Palaiahnakote, C. S. Chan, and C. L. Tan.  A robust arbitrary text detection system for natural scene images.  In Expert Systems with Applications, page 8027–8048, 2014.
  * Atienza [2021] R. Atienza.  Vision transformer for fast and efficient scene text recognition.  arXiv:2105.08582, 2021.
  * Baek et al. [2019] J. Baek, G. Kim, J. Lee, S. Park, D. Han, S. Yun, S. J. Oh, and H. Lee.  What is wrong with scene text recognition model comparisons? dataset and model analysis.  In ICCV, pages 4714–4722, 2019.
  * Borisyuk et al. [2018] F. Borisyuk, A. Gordo, and V. Sivakumar.  Rosetta: Large scale system for text detection and recognition in images.  In ACM SIGKDD, page 71–79, 2018.
  * Chen et al. [2021] J. Chen, H. Yu, J. Ma, M. Guan, X. Xu, X. Wang, S. Qu, B. Li, and X. Xue.  Benchmarking chinese text recognition: Datasets, baselines, and an empirical study.  arXiv:2112.15093, 2021.
  * Dosovitskiy et al. [2021] A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly, J. Uszkoreit, and N. Houlsby.  An image is worth 16x16 words: Transformers for image recognition at scale.  In ICLR, 2021.
  * Fang et al. [2021] S. Fang, H. Xie, Y. Wang, Z. Mao, and Y. Zhang.  Read like humans: Autonomous, bidirectional and iterative language modeling for scene text recognition.  In CVPR, 2021.
  * Gupta et al. [2016] A. Gupta, A. Vedaldi, and A. Zisserman.  Synthetic data for text localisation in natural images.  In CVPR, pages 2315–2324, 2016.
  * Hu et al. [2020] W. Hu, X. Cai, J. Hou, S. Yi, and Z. Lin.  Gtc: Guided training of ctc towards efficient and accurate scene text recognition.  In AAAI, pages 11005–11012, 2020.
  * Jaderberg et al. [2014] M. Jaderberg, K. Simonyan, A. Vedaldi, and A. Zisserman.  Synthetic data and artificial neural networks for natural scene text recognition.  In NeurIPS Deep Learning Workshop, 2014.
  * Jaderberg et al. [2015] M. Jaderberg, K. Simonyan, A. Vedaldi, and A. Zisserman.  Reading text in the wild with convolutional neural networks.  In IJCV, pages 1–20, 2015.
  * Karatzas et al. [2015] D. Karatzas, L. Gomez-Bigorda, A. Nicolaou, S. Ghosh, A. Bagdanov, M. Iwamura, J. Matas, L. Neumann, V. R. Chandrasekhar, S. Lu, F. Shafait, S. Uchida, and E. Valveny.  Icdar 2015 competition on robust reading.  In ICDAR, pages 1156–1160, 2015.
  * KaratzasAU et al. [2013] D. KaratzasAU, F. ShafaitAU, S. UchidaAU, M. IwamuraAU, L. G. i. BigordaAU, S. R. MestreAU, J. MasAU, D. F. MotaAU, J. A. AlmazànAU, and L. P. de las Heras.  Icdar 2013 robust reading competition.  In ICDAR, pages 1484–1493, 2013.
  * Li et al. [2019] H. Li, P. Wang, C. Shen, and G. Zhang.  Show, attend and read: A simple and strong baseline for irregular text recognition.  In AAAI, pages 8610–8617, 2019.
  * Liu et al. [2021] Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng Zhang, Stephen Lin, and Baining Guo.  Swin transformer: Hierarchical vision transformer using shifted windows.  In ICCV, 2021.
  * Luo et al. [2019] C. Luo, L. Jin, and Z. Sun.  A multi-object rectified attention network for scene text recognition.  In Pattern Recognit., pages 109–118, 2019.
  * Mishra et al. [2012] A. Mishra, A. Karteek, and C. V. Jawahar.  Scene text recognition using higher order language priors.  In BMVC, 2012.
  * Phan et al. [2013] T. Q. Phan, P. Shivakumara, S. Tian, and C. L. Tan.  Recognizing text with perspective distortion in natural scenes.  In CVPR, pages 569–576, 2013.
  * Sheng et al. [2019] F. Sheng, Z. Chen, and B. Xu.  Nrtr: A no-recurrence sequence-to-sequence model for scene text recognition.  In ICDAR, pages 781–786, 2019.
  * Shi et al. [2017] B. Shi, X. Bai, and C. Yao.  An end-to-end trainable neural network for image-based sequence recognition and its application to scene text recognition.  In TPAMI, pages 2298–2304, 2017.
  * Shi et al. [2019] B. Shi, M. Yang, X. Wang, P. Lyu, C. Yao, and X. Bai.  Aster: An attentional scene text recognizer with flexible rectification.  In TPAMI, pages 2035–2048, 2019.
  * Tang et al. [2022] X. Tang, Y. Lai, Y. Liu, Y. Fu, and R. Fang.  Visual-semantic transformer for scene text recognition.  In AAAI, 2022.
  * Wang et al. [2011] K. Wang, B. Babenko, and S. Belongie.  End-to-end scene text recognition.  In ICCV, pages 1457–1464, 2011.
  * Wang et al. [2021] Y. Wang, H. Xie, S. Fang, J. Wang, S. Zhu, and Y. Zhang.  From two to one: A new scene text recognizer with visual language modeling network.  In ICCV, pages 14194–14203, 2021.
  * Yan et al. [2021] R. Yan, L. Peng, S. Xiao, and G. Yao.  Primitive representation learning for scene text recognition.  In CVPR, pages 284–293, 2021.
  * Yu et al. [2020] D. Yu, X. Li, C. Zhang, T. Liu, J. Han, J. Liu, and E. Ding.  Towards accurate scene text recognition with semantic reasoning networks.  In CVPR, pages 12113–12122, 2020.
  * Zhai et al. [2016] C. Zhai, Z. Chen, J. Li, and B. Xu.  Chinese image text recognition with blstm-ctc: A segmentation-free method.  In CCPR, pages 525–536, 2016.
  * Zhang et al. [2020] H. Zhang, Q. Yao, M. Yang, Y. Xu, and X. Bai.  Autostr: Efficient backbone search for scene text recognition.  In ECCV, pages 751–767. Springer, 2020.
  * Zheng et al. [2021] T. Zheng, Z. Chen, S. Fang, H. Xie, and Y. Jiang.  Cdistnet: Perceiving multi-domain character distance for robust text recognition.  arXiv:2111.11011, 2021.


# Simplified Hartree-Fock Computations on Second-Row Atoms

S. M. Blinder
Wolfram Research, Inc.
Champaign, IL 61820, USA

Modern computational quantum chemistry has developed to a large extent
beginning with applications of the Hartree-Fock method to atoms and
molecules[1]. Density-functional methods have now largely supplanted
conventional Hartree-Fock computations in current applications of
computational chemistry. Still, from a pedagogical point of view, an
understanding of the original methods remains a necessary preliminary.
However, since more elaborate Hartree-Fock computations are now no longer
needed, it will suffice to consider a computationally simplified application
of the method[2]. Accordingly, we will represent the many-electron
wavefunction by a single closed-shell Slater determinant and use basis
functions which are simple Slater-type orbitals. This is in contrast to the
more elaborate Hartree-Fock computations, which might use multi-determinant
wavefunctions and double-zeta basis functions. All of the symbolic and
numerical operations were carried out using Mathematica.

We consider Hartree-Fock computations on the ground states of the second-row
atoms He through Ne, $Z=2$ to 10, using the simplest set of orthonormalized
$1s$, $2s$ and $2p$ orbital functions:

$\displaystyle\psi_{1s}=\frac{\alpha^{3/2}}{\sqrt{\pi}}e^{-\alpha
r},\qquad\psi_{2s}=\sqrt{\frac{3\beta^{5}}{\pi(\alpha^{2}-\alpha\beta+\beta^{2})}}\left(1-\frac{\alpha+\beta}{3}r\right)e^{-\beta
r},$
$\displaystyle\psi_{2p\\{x,y,z\\}}=\frac{\gamma^{5/2}}{\sqrt{\pi}}re^{-\gamma
r}\\{\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta\\}.\qquad$ (1)

An orbital function multiplied by a spin function $\alpha$ or $\beta$ is known
as a spinorbital, a product of the form

$\phi(x)=\psi({\bf r})\begin{cases}\alpha&\\\ \beta&\end{cases}.$ (2)

A simple representation of a many-electron atom is given by a Slater
determinant constructed from $N$ occupied spin-orbitals:

$\Psi(1,\dots,N)=\frac{1}{\sqrt{N!}}\begin{vmatrix}\phi_{a}(1)&\phi_{b}(1)&\dots&\phi_{n}(1)\\\
\phi_{a}(2)&\phi_{b}(2)&\dots&\phi_{n}(2)\\\ \vdots&\vdots&\ddots&\vdots\\\
\phi_{a}(N)&\phi_{b}(N)&\dots&\phi_{n}(N)\end{vmatrix},$ (3)

where the subscripts $a,b,\dots$ stand for spinorbitals, for example,
$1s\alpha,2p_{x}\beta$, etc. and the arguments 1, 2, $\dots$ are abbreviations
for $x_{1},x_{2},\dots$.

The Hamiltonian for an $N$-electron atom, in hartree atomic units, is given
by:

$H=\sum_{i=1}^{N}\left\\{-\frac{1}{2}\nabla_{i}^{2}-\frac{Z}{r_{i}}\right\\}+\sum_{i>j}^{N}\frac{1}{r_{ij}}.$
(4)

The corresponding approximation to the total energy is then given by

$E=\langle\Psi|H|\Psi\rangle=\sum_{i=1}^{N}H_{i}+\sum_{i>j}^{N}(J_{ij}-K_{ij}),$
(5)

where $H$, $J$ and $K$ are, respectively, the core, Coulomb and exchange
integrals:

$H_{i}=\int d^{3}{\bf r}\ \psi_{i}^{*}({\bf
r})\left\\{-\frac{1}{2}\nabla^{2}-\frac{Z}{r}\right\\}\psi_{i}({\bf r}),$ (6)
$J_{ij}=\int\int d^{3}{\bf r}d^{3}{\bf r^{\prime}}\ \frac{|\psi_{i}({\bf
r})|^{2}|\psi_{j}({\bf r^{\prime}})|^{2}}{|{\bf r}-{\bf r^{\prime}}|},$ (7)
$K_{ij}=\int\int d^{3}{\bf r}\ d^{3}{\bf r^{\prime}}\ \psi_{i}^{*}({\bf
r})\psi_{j}^{*}({\bf r^{\prime}})\frac{1}{|{\bf r}-{\bf
r^{\prime}}|}\psi_{i}({\bf r^{\prime}})\psi_{j}({\bf
r})\langle\sigma_{i}|\sigma_{j}\rangle.$ (8)

The exchange integral vanishes if $\sigma_{i}\neq\sigma_{j}$, in other words,
if spinorbitals $i$ and $j$ have opposite spins, $\alpha,\beta$ or
$\beta,\alpha$.

Explicit formulas for these integrals are given in the Appendix. Let us
illustrate with a couple of examples. The core integral $H_{2p}$ is found from

$\int_{0}^{\infty}\int_{0}^{\pi}\int_{0}^{2\pi}\frac{\gamma^{5/2}}{\sqrt{\pi}}re^{-\gamma
r}\cos\theta\Bigg{[}\left\\{-\frac{1}{2}\nabla^{2}-\frac{Z}{r}\right\\}\frac{\gamma^{5/2}}{\sqrt{\pi}}re^{-\gamma
r}\cos\theta\Bigg{]}\,r^{2}\sin\theta\,dr\,d\theta\,d\phi,$ (9)

using the Laplacian in spherical polar coordinates

$\nabla^{2}=\frac{1}{r^{2}}\frac{\partial}{\partial
r}r^{2}\frac{\partial}{\partial
r}+\frac{1}{r^{2}\sin\theta}\frac{\partial}{\partial\theta}\sin\theta\frac{\partial}{\partial\theta}+\frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}.$
(10)

The symbolic computation can be carried out using Mathematica, giving the
result

$H_{2p}=\frac{1}{2}(\gamma^{2}-Z\gamma).$ (11)

The Coulomb integral $J_{1s,2p}$ is found from

$(2\pi)^{2}\int_{0}^{\infty}\int_{0}^{\pi}\int_{0}^{\infty}\int_{0}^{\pi}\,\psi_{1s}(r_{1})^{2}\,\frac{1}{r_{12}}\psi_{2pz}(r_{2},\theta_{2})^{2}\,r_{1}^{2}\sin\theta_{1}\,dr_{1}\,d\theta_{1}\,r_{2}^{2}\sin\theta_{2}\,dr_{2}\,d\theta_{2}$
(12)

The interelectronic potential can be expanded in a series of Legendre
polynomials:

$\frac{1}{r_{12}}=\sum_{l=0}^{\infty}\frac{r_{<}^{l}}{r_{>}^{l+1}}P_{l}(\cos\Theta_{12}),$
(13)

where $r_{>}$ and $r_{<}$ are the greater and lesser of $r_{1}$ and $r_{2}$.
In terms of individual particle coordinates

$P_{l}(\cos\Theta_{12})=\frac{4\pi}{2l+1}\sum_{m=-l}^{l}Y_{l}^{m}(\theta_{1},\phi_{1})^{*}\,Y_{l}^{m}(\theta_{2},\phi_{2}).$
(14)

In the integral (12), only the $l=0$ term contributes. We must consider the
contributions both with $r_{2}>r_{1}$ and $r_{1}>r_{2}$. Using Mathematica,
the integral evaluates to

$J_{1s,2p}=\frac{\alpha\gamma(\alpha^{4}+5\alpha^{3}\gamma+10\alpha^{2}\gamma^{2}+10\alpha\gamma^{3}+2\gamma^{4})}{2(\alpha+\gamma)^{5}}.$
(15)

The energy $E$ is optimized by seeking a minimum as a function of the orbital
parameters $\alpha$, $\beta$ and $\gamma$. For example, for $Z=6$, the carbon
atom with ground state configuration $1s^{2}2s^{2}2p\,2p^{\prime}\ {}^{3}{\rm
P}_{0}$, we have

$\displaystyle
E(\alpha,\beta,\gamma)=2H_{1s}+2H_{2s}+2H_{2p}+J_{1s,1s}+J_{2s,2s}+4J_{1s,2s}-2K_{1s,2s}+$
$\displaystyle
4J_{1s,2p}-2K_{1s,2p}+4J_{2s,2p}-2K_{2s,2p}+J_{2p,2p^{\prime}}-K_{2p,2p^{\prime}}$
(16)

Following is a tabulation of the results, showing the optimal parameters and
the calculated energies. Energies are expressed in atomic units. The
computations made use of the FindMinimum function in Mathematica. Each
computation took approximately 0.005 seconds of CPU time (using a MacBook Pro
with the Apple M1 chip). For comparison, we also include the results of the
best Hartree-Fock computations[3] and the exact nonrelativistic energies of
the atomic ground states. Our approximate energies are within 1% of the
accurate H-F values, obtained with an order of magnitude less computational
effort.

##  Hartree-Fock Results

$Z$ | atom | configuration | $\alpha$ | $\beta$ | $\gamma$ | calculated $E$ | best H-F | exact
---|---|---|---|---|---|---|---|---
2 | He |  $1s^{2}\ {}^{1}$S | 1.6875 |  |  | -2.84766 | -2.86168 | -2.903385
3 | Li |  $1s^{2}2s\ {}^{2}$S | 2.69372 | 0.766676 |  | \- 7.41385 | -7.43273 | -7.477976
4 | Be |  $1s^{2}2s^{2}\ {}^{1}$S | 3.70767 | 1.15954 |  | \- 14.5300 | -14.5730 | -14.668449
5 | B |  $1s^{2}2s^{2}2p\ {}^{2}$P | 4.71099 | 1.57921 | 1.18716 | -24.4506 | -24.5291 | -24.658211
6 | C |  $1s^{2}2s^{2}2p^{2}\ {}^{3}$P | 5.71244 | 1.98775 | 1.51874 | -37.5471 | -37.5471 | -37.855668
7 | N |  $1s^{2}2s^{2}2p^{3}\ {}^{4}$S | 6.71293 | 2.39148 | 1.84551 | -53.9624 | -54.4009 | -54.611893
8 | O |  $1s^{2}2s^{2}2p^{4}\ {}^{3}$P | 7.71286 | 2.79267 | 2.16972 | -74.1624 | -74.8094 | -75.109991
9 | F |  $1s^{2}2s^{2}2p^{5}\ {}^{2}$P | 8.71243 | 3.19236 | 2.49238 | -98.3972 | -99.4093 | -99.803888
10 | Ne |  $1s^{2}2s^{2}2p^{6}\ {}^{1}$S | 9.71176 | 3.59108 | 2.81404 | -126.971 | -128.547 | -128.830462

##  Appendix: Core, Coulomb and Exchange Integrals

## References

[1] S. M. Blinder, “Introduction to the Hartree-Fock Method,” in Mathematical
Physics in Theoretical Chemistry (S. M. Blinder and J. E. House, eds.),
Elsevier, 2018.

[2] This paper is based in the Wolfram Demonstration:
https://demonstrations.wolfram.com/SimplifiedHartreeFockComputationsOnSecondRowAtoms/

[3] Atomic Data and Nuclear Data Tables, 95(6), 2009 pp. 836-870.


# On the distribution of sequences of the form $(q_{n}y)$

S. Kristensen  S. Kristensen, Department of Mathematics, Aarhus University, Ny
Munkegade 118, DK-8000 Aarhus C, Denmark<EMAIL_ADDRESS>and  T. Persson  T.
Persson, Centre for Mathematical Sciences, Lund University, Box 118, SE-22100
Lund, Sweden<EMAIL_ADDRESS>
###### Abstract.

We study the distribution of sequences of the form $(q_{n}y)_{n=1}^{\infty}$,
where $(q_{n})_{n=1}^{\infty}$ is some increasing sequence of integers. In
particular, we study the Lebesgue measure and find bounds on the Hausdorff
dimension of the set of points $\gamma\in[0,1)$ which are well approximated by
points in the sequence $(q_{n}y)_{n=1}^{\infty}$. The bounds on Hausdorff
dimension are valid for almost every $y$ in the support of a measure of
positive Fourier dimension. When the required rate of approximation is very
good or if our sequence is sufficiently rapidly growing, our dimension bounds
are sharp. If the measure of positive Fourier dimension is itself Lebesgue
measure, our measure bounds are also sharp for a very large class of
sequences. We also give an application to inhomogeneous Littlewood type
problems.

###### 2020 Mathematics Subject Classification:

11J83, 28A78

We thank Niclas Technau for pointing out to us that the paper [2] contains a
result that implies a previous version of our Threorem 1. This lead to the
present and stronger version of Threorem 1. SK’s research is supported by the
Independent Research Fund Denmark (Grant ref. 1026-00081B) and the Aarhus
University Research Foundation (Grant ref. AUFF-E-2021-9-20)

##  1\. Introduction

Let $y$ be fixed real number and let $(q_{n})_{n=1}^{\infty}$ be a sequence of
positive integers. We consider the set

(1) $W_{y,\alpha}=\\{\,\gamma\in[0,1):\|q_{n}y-\gamma\|<n^{-\alpha}\text{ for
infinitely many }n\in\mathbb{N}\,\\},$

where $\|\cdot\|$ denotes the distance to the nearest integer.

The study of this set is not a new one. For instance, if
$(q_{n})_{n=1}^{\infty}$ grows exponentially but not super-exponentially fast
and $\alpha<\frac{1}{2}$, it is shown i [7] that for almost every $y$ with
respect to any measure with sufficiently nice Fourier transform, the set
$W_{y,\alpha}$ is in fact the entire interval $[0,1)$.

The motivation in [7] for studying this set was applications to certain
Littlewood type problems. More concretely, a version of the inhomogeneous
Littlewood conjecture states that for all $x$, $y$ and $\gamma$,

$\liminf_{q\rightarrow\infty}q\|qx\|\|qy-\gamma\|=0,$

where the liminf is over all positive integers. A consequence of the main
result in [7] is that for any fixed badly approximable number
$x\in\operatorname{Bad}$, there is a set $G\subseteq\operatorname{Bad}$ of
maximal Hausdorff dimension such that for all $y\in G$ and all $\gamma$,

(2) $q\|qx\|\|qy-\gamma\|<\frac{1}{(\log q)^{1/2-\epsilon}}.$

As always, the set $\operatorname{Bad}$ is the set of badly approximable
numbers, i.e.

$\operatorname{Bad}=\Bigl{\\{}\,x\in[0,1):\liminf_{q\rightarrow\infty}q\|qx\|=0\,\Bigr{\\}}.$

Of course, this set is equal to the set of numbers for which the sequence of
partial quotients in the simple continued fraction expansion is bounded, see
e.g. [9].

In fact, this was shown to hold for $\mu$-almost every $y$ in the support of a
probability measure $\mu$ of positive Fourier dimension, a notion which we
will define along with the notion of Hausdorff dimension in the next section.

In fact, the right hand side of (2) was improved by Chow and Zafeiropoulos [3]
to $(\log\log\log q)^{\epsilon+1/2}/(\log q)^{1/2}$, though one should note
that the power of the logarithm in the denominator is $1/2$. This reflects in
the exponent $\alpha$ in the definition of the set $W_{y,\alpha}$, and it
appears to be very difficult to prove properties of this set for $\alpha\geq
1/2$ as remarked in [12]. In fact, it follows from the work of Technau and
Zafeiropoulos [14] that methods involving the discrepancy of sequences of the
form $(q_{n}\alpha)_{n=1}^{\infty}$ will not accomplish this.

We are able to obtain some results for $\alpha$ in the range between $1/2$ and
$1$, although we suspect that these are not best possible. For some non-
lacunary (and for lacunary) sequences $(q_{n})$ we are able to prove that for
$\mu$-almost every $y$ in the support of a probability measure of positive
Fourier dimension, the set $W_{y,\alpha}$ has positive Lebesgue measure. If
the sequence $(q_{n})$ is lacunary, then Chow and Technau [2] showed that
$W_{y,\alpha}=[0,1)$ for $\mu$-almost every $y$.

If $\mu$ is the Lebesgue measure, we are able to relax the growth condition on
the sequence $(q_{n})$ to being just a little more than linear, while at the
same time prove that the Lebesgue measure of $W_{y,\alpha}$ is equal to $1$.
Of course, the result of Chow and Technau implies this and more when $(q_{n})$
is lacunary, but our result allows for less restrictive growth conditions.

For $\alpha>1$, an application of the Borel–Cantelli lemma immediately shows
that the set is Lebesgue null for all $x$. However, it is not usually empty,
and in this note we will derive some properties of the set $W_{y,\alpha}$ in
the $\alpha>1$ range. A special case of our results is that for Lebesgue
almost all $y$, the set has Hausdorff dimension $\frac{1}{\alpha}$, though our
result is more general. We will state it precisely, once we have provided the
technical background for doing so.

In the above mentioned result on the Hausdorff dimension, it is critical that
the sequence $(q_{n})_{n=1}^{\infty}$ grows geometrically fast. Nonetheless,
the problem of studying the distribution of orbits of fractional parts of
$q_{n}y$ for such sequences has attracted quite a bit of attention. For
instance, if the sequence $(q_{n})_{n=1}^{\infty}$ contains all numbers of the
form $2^{i}3^{j}$ in increasing order, Furstenberg [6] famously showed that
orbits are either finite or dense and conjectured that dense orbits are
uniformly distributed. Similarly, initiated by the work of Rudnick and Sarnak
[13], the distribution of sequences of the form $(n^{d}y)_{n=1}^{\infty}$ has
attracted a considerable amount of attention. While we do not claim to solve
any of the famous and important open problems about the distribution of these
sequences, we are able to find upper and lower bounds on the Hausdorff
dimension of $W_{y,\alpha}$ regardless of any growth conditions, which are
valid for almost all $y$ in an appropriate sense. Specialising our main result
to the Lebesgue measure, we find that for $\alpha>1$ and for Lebesgue almost
every $y$, the Hausdorff dimension of $W_{y,\alpha}$ is somewhere in between
$\frac{1}{2\alpha}$ and $\frac{1}{\alpha}$.

The paper is structured as follows. In the next section, we will provide the
relevant definitions of notions from uniform distribution theory and fractal
geometry. With these notions in place, we will state our main theorems in
Section 3, where we will also derive some corollaries of these. The main
theorems are proved in Section 4. We end the paper with some concluding
remarks.

##  2\. Some technical background

We will let $\lambda$ denote the Lebesgue measure on $[0,1]$. Our approach
depends heavily on the notion of the discrepancy of a sequence, which is a
measure of how far the sequence is from being uniformly distributed. For a
sequence $(x_{n})_{n=1}^{\infty}$ in $[0,1)$ and $N\in\mathbb{N}$, we define
the $N$-discrepancy of the sequence by

$D_{N}(x_{n})=\sup_{I\subseteq[0,1)}|\\#\\{\,n\leq N:x_{n}\in
I\,\\}-N\,\lambda(I)|,$

where the _supremum_ is taken over all sub-intervals of $[0,1)$. The sequence
$(x_{n})_{n=1}^{\infty}$ is said to be uniformly distributed in $[0,1)$ if
$D_{N}(x_{n})=o(N)$.

A word of caution is appropriate here. Some authors define the discrepancy of
a sequence to be the above quantity divided by $N$. In this notion of
discrepancy, the quantity can perhaps most easily be understood to be a speed
of convergence of the probability measures
$\frac{1}{N}\sum_{n=1}^{N}\delta_{x_{n}}$ to the Lebesgue measure. The former
notion favours counting over measure theory. The two notions are of course
entirely equivalent, but the double meaning of the word ‘discrepancy’ can
cause confusion. _In this paper, discrepancy is taken to mean the former._

Many of our results deal with Hausdorff dimension, which we now define. For a
set $E\subseteq\mathbb{R}^{n}$ and real numbers $s\geq 0$ and $\delta>0$,
consider the quantity,

$\mathcal{H}_{\delta}^{s}(E)=\inf_{\mathcal{C}_{\delta}}\sum_{U\in\mathcal{C_{\delta}}}\operatorname{diam}(U)^{s},$

where the _infimum_ is taken over all covers $\mathcal{C}_{\delta}$ of $E$
with sets of diameter at most $\delta$. As $\delta$ decreases, the collection
of such covers becomes smaller, so that $\mathcal{H}_{\delta}^{s}(E)$ must
increase. Allowing infinite limits, this means that

$\mathcal{H}^{s}(E)=\lim_{\delta\rightarrow 0}\mathcal{H}_{\delta}^{s}(E)$

exists. In fact, it is a regular Borel measure, and for all but possibly one
value of $s$, the value attained by $\mathcal{H}^{s}(E)$ is either $0$ or
$\infty$. Furthermore, $\mathcal{H}^{s}(E)=0$ for $s>n$. Since
$\mathcal{H}^{s}(E)$ evidently decreases as $s$ increases, we can define the
Hausdorff dimension of $E$ to be the ‘breaking point’, i.e.

$\operatorname{dim_{H}}(E)=\inf\\{\,s\geq 0:\mathcal{H}^{s}(E)=0\,\\}.$

A particularly pleasing type of sets are intersective sets in the sense of
Falconer, see [5]. For $s>0$, the Falconer class $\mathcal{G}^{s}$ is defined
as the collection of $G_{\delta}$-sets $F$ of $\mathbb{R}^{n}$ for which

$\operatorname{dim_{H}}\biggl{(}\bigcap_{i=1}^{\infty}f_{i}(F)\biggr{)}\geq
s,$

for any countable sequence $(f_{i})_{i=1}^{\infty}$ of similarities. Among the
desirable properties of the families $\mathcal{G}^{s}$ we find the fact that
it is closed under countable intersections.

We will consider the intersection classes $\mathcal{G}^{s}$ restricted to
$[0,1)$, which we denote by $\mathcal{G}^{s}([0,1))$. A set $E$ belongs to
$\mathcal{G}^{s}([0,1))$ if there is a set $F\in\mathcal{G}^{s}$ such that
$E=[0,1)\cap F$. Sets in $\mathcal{G}^{s}([0,1))$ have Hausdorff dimension at
least $s$, and $\mathcal{G}^{s}([0,1))$ is closed under countable
intersections.

A final notion, which will play an important role is the Fourier dimension of
a measure, see e.g. [10]. The Fourier dimension of a Radon measure $\mu$ on
$\mathbb{R}$ is defined as

(3) $\dim_{F}(\mu)=\sup\\{\,s\in[0,1]:|\hat{\mu}(\xi)|=O(|\xi|^{-s/2})\,\\}.$

Here, $\hat{\mu}$ is the Fourier transform of the measure defined as usual to
be

$\hat{\mu}(\xi)=\int e^{2\pi ix\xi}\,\mathrm{d}\mu(x).$

The terminology originates in the fact that for a set $E\subseteq\mathbb{R}$,
any measure supported on $E$ has Fourier dimension at most equal to the
Hausdorff dimension of $E$. The Fourier dimension of the set $E$ is the
supremum of the Fourier dimensions of the Radon measures supported on $E$.

Note that a set $E$ need not have the same Hausdorff and Fourier dimension.
For instance, the ternary Cantor set has Hausdorff dimension $\log 2/\log 3$
by a classical calculation found in all textbooks on fractal geometry.
However, it was shown by Kahane and Salem [8] that the Fourier dimension of
the ternary Cantor set is equal to zero. This result in fact follows rather
easily from the results of both [7] and this paper.

##  3\. Statement of results

From [7], we know that if $\mu$ is some measure supported on $[0,1)$ of
positive Fourier dimension, then for $\mu$-almost every $x$, there is a
positive number $\nu$, such that $W_{y,\alpha}=[0,1]$ for $\alpha<\nu$. If we
further assume that $(q_{n})_{n=1}^{\infty}$ is lacunary, then Chow and
Technau [2, Theorem 1.11] recently showed that this holds true for any
$\alpha<1$. In fact they obtained an even stronger result: Changing the rate
$n^{-\alpha}$ to $n^{-1}(\log n)^{3+\varepsilon}$, they proved that

$\\{\,\gamma\in[0,1):\|q_{n}y-\gamma\|<n^{-1}(\log n)^{3+\varepsilon}\text{
for infinitely many }n\in\mathbb{N}\,\\}=[0,1)$

for almost every $y$.

Our first results is valid also for some non-lacunary sequences including
sequences of the form $q_{n}=n^{d}$.

######  Theorem 1.

Suppose that $\mu$ is a probability measure of positive Fourier dimension with
decay $|\hat{\mu}(\xi)|=O(|\xi|^{-\tau})$. If $(q_{n})_{n=1}^{\infty}$ is a
sequence of integers with $|q_{n}-q_{m}|>c|n-m|^{\frac{1}{\tau}+\varepsilon}$
for some $c,\varepsilon>0$, and $\alpha<1$, then $\lambda(W_{y,\alpha})>0$ for
$\mu$-almost every $y$.

In particular, Theorem 1 applies to the case when $\mu$ is the Lebesgue
measure and $(q_{n})$ is given by an integer polynomial of degree at least
$3$.

Specialising $\mu$ to be the Lebesgue measure on $[0,1]$, we can obtain the
following stronger result.

######  Theorem 2.

Suppose that $(q_{n})_{n=1}^{\infty}$ is a sequence of integers with
$|q_{n}-q_{m}|>c|n-m|^{1+\varepsilon}$. For $\alpha<1$, the set $W_{y,\alpha}$
is Lebesgue full for Lebesgue-almost every $y$.

We should remark that the growth condition in Theorem 1 can be relaxed to
$|q_{n}-q_{m}|>c|n-m|^{\frac{1}{\tau}}(\log|n-m|)^{1+\varepsilon}$ and
similarly in Theorem 2. As it will be apparent in the proofs, all that matters
is that a certain series is convergent.

We are not able to push the results for $\alpha<1$ any further. However as
already mentioned, for $\alpha>1$ the first Borel–Cantelli lemma immediately
implies that the set $W_{y,\alpha}$ is a Lebesgue null set, and it is of
interest to calculate its Hausdorff dimension. We are able to get a precise
result valid almost surely in the case of lacunary sequences.

######  Theorem 3.

Let $\mu$ be a probability measure on $[0,1]$ with positive Fourier dimension,
and let $(q_{n})_{n=1}^{\infty}$ be a lacunary sequence of integers. For any
$\alpha\geq 1$, for $\mu$-almost all $x$,

$\operatorname{dim_{H}}(W_{y,\alpha})=\frac{1}{\alpha}.$

In fact, the set is intersective in the sense of Falconer.

As in [7], our result has applications to a Littlewood type question, where
one adds an inhomogeneous term. More precisely, we can show the following.

######  Theorem 4.

Let $\mu$ be a probability measure on $[0,1]$ with positive Fourier dimension,
let $x\in\operatorname{Bad}$, and let $m\in\mathbb{N}$. Then, for
$\mu\times\ldots\times\mu$-almost all $(y_{1},\ldots,y_{m})\in[0,1)^{m}$,

$\operatorname{dim_{H}}\biggl{\\{}\,\gamma\in[0,1):q\|qx\|\prod_{i=1}^{m}\|qy_{i}-\gamma\|<\frac{1}{(\log
q)^{m\alpha}}\text{ infinitely often}\,\biggr{\\}}=\frac{1}{\alpha}.$

As in [7], one could extend this to require that there are infinitely many
solutions to these inequalities for a countable family of badly approximable
$x_{j}$’s. Also, just as in that paper, replacing $\|qx\|$ by the $p$-adic
absolute value of $q$, $|q|_{p}$, or more generally by the pseudo-absolute
value $|q|_{\mathcal{D}}$, where $\mathcal{D}=(d_{k})_{k=1}^{\infty}$ is a
sequence of positive integers with $d_{k}|d_{k+1}$ of at most geometric growth
yields the same result. In that case, one considers an inhomogeneous analogue
of the mixed Littlewood conjecture.

Let us derive Theorem 4 from Theorem 3.

###### Proof of Theorem 4.

First, let $(q_{n})_{n=1}^{\infty}$ denote the sequence of denominators of
convergents to $x$. From Theorem 3, we know that for $\mu$-almost every
$y_{1}$, the set $W_{y_{1},\alpha}$ is intersective of dimension $1/\alpha$.
Now, extract from the sequence $(q_{n})_{n=1}^{\infty}$ a subsequence
$(q_{n_{j}})_{j=1}^{\infty}$ with

$\|q_{n_{j}}y_{1}-\gamma\|<n_{j}^{-\alpha}.$

The subsequence is again lacunary, so we will use Theorem 3. Hence, for
$\mu$-almost every $y_{2}$, the set $W_{y_{2},\alpha}$ is intersective of
dimension $1/\alpha$ with this new sequence.

We proceed in this way $m$ times, until we have a sequence, also denoted
$(q_{n_{j}})_{j=1}^{\infty}$ by abuse of notation, such that for
$i=1,2,\ldots,m$,

(4) $\|q_{n_{j}}y_{i}-\gamma\|<n_{j}^{-\alpha},$

whenever $\gamma\in W_{y_{i},\alpha}$. Thus, (4) holds for every $i$, provided
$\gamma\in\bigcap_{i=1}^{m}W_{y_{i},\alpha}$. Since these sets are in
Falconer’s class of intersective sets, the dimension of the intersection
remains $1/\alpha$. Multiplying everything, and noting that
$q_{n_{j}}\|q_{n_{j}}x\|\leq 1$,

$q_{n_{j}}\|q_{n_{j}}x\|\prod_{i=1}\|q_{n_{j}}y_{i}-\gamma\|<\frac{1}{n_{j}^{m\alpha}}.$

To conclude, we only need to know that the sequence of denominators of
convergents of a badly approximable number satisfies the inequalities
$\phi^{n}\leq q_{n}\leq(2M)^{n}$, where $M$ is an upper bound on the partial
quotients of the number and $\phi$ is the Golden Ratio. Thus $\log
q_{n_{j}}\asymp n_{j}$, and we are done. ∎

To obtain a version of our result for inhomogeneous mixed Littlewood type
problems, one just needs to replace the sequence of denominators of
convergents by the sequence $\mathcal{D}$. The details are left to the
interested reader.

For non-lacunary sequences $(q_{n})_{n=1}^{\infty}$, our results are less
precise. Nonetheless, the problem of studying the the distribution of orbits
of fractional parts of $q_{n}y$ for such sequences has attracted quite a bit
of attention. For instance, if the sequence $(q_{n})_{n=1}^{\infty}$ contains
all numbers of the form $2^{i}3^{j}$ in increasing order, Furstenberg [6]
famously showed that orbits are either finite or dense and conjectured that
dense orbits are uniformly distributed. Similarly, initiated by the work of
Rudnick and Sarnak [13], the distribution of sequences of the form
$(n^{d}y)_{n=1}^{\infty}$ has attracted a considerable amount of attention.
While we do not claim to solve any of the famous and important open problems
about the distribution of these sequences, we will prove the following.

######  Theorem 5.

Let $(x_{n})_{n=1}^{\infty}$ be a sequence in $[0,1)$ with
$D_{N}((x_{n}))=O(N^{1-\eta})$ for some $\eta\in(0,1)$. Then the set

$W=\\{\,\gamma\in[0,1):\|x_{n}-\gamma\|<n^{-\alpha}\text{ infinitely
often}\,\\}$

satisfies that
$\frac{\eta}{\alpha}\leq\operatorname{dim_{H}}W\leq\frac{1}{\alpha}$ for
$\alpha\geq 1$.

In [7], it is shown that for an increasing sequence of integers
$(q_{n})_{n=1}^{\infty}$, the sequence $(\\{q_{n}y\\})_{n=1}^{\infty}$
satisfies the required discrepancy estimate for $\mu$-almost every $y$,
whenever $\mu$ is a measure of positive Fourier dimension. More precisely, it
follows from the proof of Corollary 7 in [7] that if
$|\hat{\mu}(\xi)|=O(|\xi|^{-2\eta})$, then
$D_{N}((x_{n}))=O(N^{1-\eta^{\prime}})$ for any $\eta^{\prime}<\eta$. Thus, we
immediately have the following corollary.

######  Corollary 6.

Let $\mu$ be a probability measure on $[0,1]$ with Fourier decay
$|\hat{\mu}(\xi)|=O(|\xi|^{-2\eta})$, and let $(q_{n})_{n=1}^{\infty}$ be an
increasing sequence of integers. For any $\alpha\geq 1$, for $\mu$-almost all
$y$,

$\frac{\eta}{\alpha}\leq\operatorname{dim_{H}}W_{y,\alpha}\leq\frac{1}{\alpha}.$

Note in particular that although the Fourier dimension of the Lebesgue measure
$\lambda$ is 1, we have the stronger Fourier decay
$|\hat{\lambda}(\xi)|=O(|\xi|^{-1})$. (This is because in the definition (3)
of the Fourier dimension, the supremum is taken over the interval $[0,1]$.)
Hence, in the case of Lebesgue measure, Corollary 6 implies that
$\frac{1}{2\alpha}\leq\operatorname{dim_{H}}W_{y,\alpha}\leq\frac{1}{\alpha}$
for Lebesgue almost every $y$.

Theorem 5 does not give a precise Hausdorff dimension of the set $W$, but
under an additional assumption on the sequence $(x_{n})_{n=1}^{\infty}$, we do
in fact get the exact dimension. If we define the quantity,

$d_{N}((x_{n}))=\inf_{1\leq k<l\leq N}\lVert x_{k}-x_{l}\rVert,$

we are able to prove the following.

######  Theorem 7.

Let $(x_{n})$ be a sequence in $[0,1)$ with $D_{N}((x_{n}))\leq cN^{1-\eta}$
and $d_{N}((x_{n}))\geq cN^{-\beta}$, where $0<\eta<1$ and
$1\leq\beta\leq\eta(\alpha-1)+1$. Put

$W=\\{\,y\in[0,1):\lVert x_{n}-y\rVert<n^{-\alpha}\text{ infinitely
often}\,\\}.$

Then $\operatorname{dim_{H}}W=\frac{1}{\alpha}$.

We get the following corollary to Theorem 7.

######  Corollary 8.

Let $(q_{n})_{n=1}^{\infty}$ be an increasing sequence of integers. For any
$\alpha\geq 3$ and $\lambda$-almost every $y$, we have

$\operatorname{dim_{H}}W_{y,\alpha}=\frac{1}{\alpha}.$

##  4\. Proofs

We first prove Theorem 1.

###### Proof of Theorem 1.

We put

$A_{k}=A_{k}(y)=\\{\,\gamma\in[0,1):\lVert
q_{k}y-\gamma\rVert<k^{-\alpha}\,\\}=B(q_{k}y,k^{-\alpha}).$

Then

$\lambda(A_{k})=2k^{-\alpha}.$

We also have for $k<l$ that

(5) $\lambda(A_{k}\cap A_{l})\leq
2l^{-\alpha}\mathbbm{1}_{B(0,2k^{-\alpha})}((q_{k}-q_{l})y)$

since $\lambda(A_{k}\cap A_{l})\leq\lambda(A_{l})=2l^{-\alpha}$ always holds
and $A_{k}\cap A_{l}=\emptyset$ holds when $(q_{k}y-q_{l}y)\not\in
B(0,2k^{-\alpha})$.

We now consider the average $\int\lambda(A_{k}(y)\cap
A_{l}(y))\,\mathrm{d}\mu(y)$. Writing $\mathbbm{1}_{B(0,2k^{-\alpha})}$ as a
Fourier series,

$\mathbbm{1}_{B(0,2k^{-\alpha})}(y)=\sum_{j\in\mathbb{Z}}a_{j}e^{i2\pi jy},$

we can estimate the Fourier coefficients by $|a_{j}|\leq 1/|j|$ for $j\neq 0$
and we have $a_{0}=4k^{-\alpha}$. This implies that

$\int\mathbbm{1}_{B(0,2k^{-\alpha})}((q_{k}-q_{l})y)\,\mathrm{d}\mu(y)=\sum_{j}a_{j}\int
e^{i2\pi j(q_{k}-q_{l})y}\,\mathrm{d}\mu\leq a_{0}+\sum_{j\neq
1}j^{-1}\hat{\mu}(j(q_{k}-q_{l})).$

Since $\mu$ has positive Fourier dimension with decay rate $\tau$, and by the
assumption on the sequence $(q_{n})$, we have
$\hat{\mu}(j(q_{k}-q_{l}))\leq\tilde{c}_{1}j^{-\tau}(l-k)^{-1-\tau\varepsilon}$.
Therefore, we have

$\int\mathbbm{1}_{B(0,2k^{-\alpha})}((q_{k}-q_{l})y)\,\mathrm{d}\mu(y)\leq
4(k^{-\alpha}+c_{1}(l-k)^{-1-\tau\varepsilon})$

for some constant $c_{1}$. By (5) we then have

(6) $\int\lambda(A_{k}\cap A_{l})\,\mathrm{d}\mu\leq
8l^{-\alpha}(k^{-\alpha}+c_{1}(l-k)^{-1-\tau\varepsilon}).$

Take $m<n$ and put

$S_{m,n}(y)=S_{m,n}=\sum_{k=m}^{n}\lambda(A_{k}),\qquad C_{m,n}(y)=\sum_{m\leq
k,l\leq n}\lambda(A_{k}(y)\cap A_{l}(y)),$

We consider the sets

$\Delta_{m,n}(p)=\biggl{\\{}\,y:C_{m,n}(y)>p\int
C_{m,n}(y)\,\mathrm{d}\mu(y)\,\biggr{\\}}$

For $p\geq 1$, we have $\mu(\Delta_{m,n}(p))\leq p^{-1}$. Therefore,
$\mu(\complement\Delta_{m,n}(p))\geq 1-p^{-1}$ and

$\complement\Delta_{m,n}(p)=\biggl{\\{}\,y:C_{m,n}(y)\leq p\int
C_{m,n}(y)\,\mathrm{d}\mu(y)\,\biggr{\\}}.$

Fix $p>1$ and take a quickly increasing sequence $n_{j}$ going to infinity in
such a way that $\frac{n_{j+1}}{n_{j}}\to\infty$. Put
$\Delta_{j}(p)=\Delta_{n_{j},n_{j+1}}(p)$. Then
$G(p)=\limsup\complement\Delta_{j}(p)$ satisfies $\mu(G(p))\geq 1-p^{-1}$.

Take $y\in G(p)$. Then there are infinitely many $j$ for which

$C_{n_{j},n_{j+1}}(y)\leq pC_{j},$

where $C_{j}=\int C_{n_{j},n_{j+1}}\,\mathrm{d}\mu$.

Put $S_{j}=S_{n_{j},n_{j+1}}$ We estimate $C_{j}$ and $S_{j}$. By (6) we have

$\displaystyle C_{j}$ $\displaystyle\leq
S_{j}+2\sum_{l=n_{j}+1}^{n_{j+1}}\sum_{k=n_{j}}^{l-1}8l^{-\alpha}(k^{-\alpha}+c_{1}(l-k)^{-1-\tau\varepsilon})$
$\displaystyle\leq
S_{j}+2\sum_{l=n_{j}+1}^{n_{j+1}}8l^{-\alpha}\biggl{(}\int_{0}^{l}x^{-\alpha}\,\mathrm{d}x+c_{2}\biggr{)}$
$\displaystyle=S_{j}+16\sum_{l=n_{j}+1}^{n_{j+1}}l^{-\alpha}\biggl{(}\frac{1}{1-\alpha}l^{1-\alpha}+c_{2}\biggr{)}$
(7) $\displaystyle\leq
S_{j}+\frac{16}{(1-\alpha)^{2}}(n_{j+1}+1)^{2-2\alpha}+\frac{16c_{2}}{(1-\alpha)}(n_{j+1}+1)^{1-\alpha}.$

We also have

(8) $S_{j}=\sum_{k=n_{j}}^{n_{j+1}}2k^{-\alpha}\geq
2\int_{n_{j}}^{n_{j+1}}x^{-\alpha}\,\mathrm{d}x=\frac{2}{1-\alpha}(n_{j+1}^{1-\alpha}-n_{j}^{1-\alpha}).$

For any such $j$, we then have by the very famous and important Chung–Erdős
inequality [4] that

$\lambda\biggl{(}\bigcup_{k=n_{j}}^{n_{j+1}}A_{k}(y)\biggr{)}\geq\frac{S_{n_{j},n_{j+1}}^{2}}{C_{n_{j},n_{j+1}}}\geq\frac{S_{j}^{2}}{pC_{j}}.$

By the estimates (7) and (8) we have that

$\lim_{j\to\infty}\frac{S_{j}^{2}}{C_{j}}=\frac{1}{4p}.$

It follows that for any $y\in G(p)$ we have
$\lambda(W_{y,\alpha})\geq\frac{1}{4p}>0$. Hence $\lambda(W_{y,\alpha})>0$ for
any $y\in\bigcup_{p>1}G(p)$ and $\mu(\bigcup_{p>1}G(p))=1$. ∎

Now, we consider the special case of $\mu=\lambda$ and prove Theorem 2.

###### Proof of Theorem 2.

We first note that $|\hat{\lambda}(\xi)|=O(|\xi|^{-1})$, whence by Theorem 1,
$W_{y,\alpha}$ will have positive Lebesgue measure for any $\alpha<1$.

We consider the auxiliary set

$\overline{W}_{\alpha}=\\{\,(x,y)\in[0,1]^{2}:\|q_{n}y-x\|<n^{-\alpha}\text{
infinitely often }\,\\}.$

By Theorem 1 and Fubini’s Theorem, the two dimensional Lebesgue measure of
this set is positive for any $\alpha<1$.

Now, fix $\alpha^{\prime}\in(\alpha,1)$. By the above,
$\lambda(\overline{W}_{\alpha^{\prime}})>0$. We will use this and an inflation
argument due to Cassels [1] to prove that this implies that
$\lambda(\overline{W}_{\alpha})=1$.

Let $\epsilon>0$. We split up the square $[0,1)^{2}$ into $T^{2}$ disjoint
squares of side length $1/T$, where $T\in\mathbb{N}$ is chosen by Lemma VII.5
in [1], so that for one of these boxes, $B$ say,

$\lambda(\overline{W}_{\alpha^{\prime}}\cap B)>(1-\epsilon)\lambda(B).$

Scaling the box by $T^{2}$ and projecting back onto the $[0,1)^{2}$ by
reducing modulo $1$, we cover the entire unit square, so that

$\lambda(T(\overline{W}_{\alpha^{\prime}}\cap B)\pmod{1})>(1-\epsilon).$

Since $\epsilon>0$ was arbitrary, it follows that for every pair
$(x,y)\in[0,1)^{2}$ outside of a set of Lebesgue measure $<\epsilon$, there is
a point $(x^{\prime},y^{\prime})\in\overline{W}_{\alpha^{\prime}}$ with
$(x,y)=T(x^{\prime},y^{\prime})\pmod{1}$.

Now, for such a point,

$\|q_{n}y-x\|=\|T(q_{n}y^{\prime}-x^{\prime})\|\leq
T\|q_{n}y^{\prime}-x^{\prime}\|.$

Since $(x^{\prime},y^{\prime})\in\overline{W}_{\alpha^{\prime}}$, for
infinitely many values of $n$,

$T\|q_{n}y^{\prime}-x^{\prime}\|<Tn^{-\alpha^{\prime}}=Tn^{\alpha-\alpha^{\prime}}n^{-\alpha}.$

Since $\alpha^{\prime}>\alpha$, $Tn^{\alpha-\alpha^{\prime}}\leq 1$ for $n$
large enough, since $T$ was fixed. Consequently,
$(x,y)\in\overline{W}_{\alpha^{\prime}}$, so that
$\lambda(\overline{W}_{\alpha})>1-\epsilon$. Since $\epsilon>0$ was arbitrary,
it follows that $\lambda(\overline{W}_{\alpha})=1$.

Now, to conclude we apply Fubini’s Theorem once more and find that almost all
fibers $W_{y,\alpha}$ are Lebesgue full. ∎

We now prove Theorem 3. In the proof, we will make use of the following
quantity related to discrepancy,

$\tilde{D}_{N}(x_{n})=\sup_{I\subseteq[0,1)}|\\#\\{\,N+1\leq n\leq 2N:x_{n}\in
I\,\\}-N\,\lambda(I)|.$

We will need the following lemmata.

######  Lemma 9.

If $D_{N}((x_{n}))\leq O(N^{1-\eta})$ then $\tilde{D}_{N}((x_{n}))\leq
O(N^{1-\eta})$.

###### Proof.

Take $I\subset[0,1)$. Then

$\displaystyle\biggl{|}\sum_{n=N+1}^{2N}\mathbbm{1}_{I}(x_{n})-N|I|\biggr{|}$
$\displaystyle=\biggl{|}\sum_{n=1}^{2N}\mathbbm{1}_{I}(x_{n})-2N|I|-\sum_{n=1}^{N}\mathbbm{1}_{I}(x_{n})+N|I|\biggr{|}$
$\displaystyle\leq D_{2N}((x_{n}))+D_{N}((x_{n})).\qed$

We will also need the following result of Persson and Reeve [11].

######  Lemma 10 (Persson and Reeve [11]).

Let $(\mu_{n})$ be a sequence of probability measures with
$\operatorname{supp}\mu_{n}\subset E_{n}\subset[0,1]$, and such that
$\mu_{n}\to\lambda$ weakly. If there is a constant $C$ such that

$I_{s}(\mu_{n}):=\iint|x-y|^{-s}\,\mathrm{d}\mu(x)\mathrm{d}\mu(y)<C$

for all $n$, then the set $E=\limsup E_{n}$ has Hausdorff dimension at least
$s$, and $E$ has large intersections as defined in Section 2.

The reader will have noted that the upper bound on the dimension in each case
is the same. It is the easier part of all the statements. We prove it here as
a lemma.

######  Lemma 11.

Let $(x_{n})_{n=1}^{\infty}$ be any sequence in $[0,1)$ and let $\alpha\geq
1$. Then,

$\operatorname{dim_{H}}\\{\,\gamma\in[0,1):\|x_{n}-\gamma\|<n^{-\alpha}\text{
infinitely often}\,\\}\leq\frac{1}{\alpha}.$

###### Proof.

Let $s=1/\alpha+\epsilon$ for some $\epsilon>0$. For any $N\in\mathbb{N}$, the
set, $W$ say, in the statement of the lemma is covered by the intervals
$(x_{n}-n^{-\alpha},x_{n}+n^{-\alpha})$, each of length $2n^{-\alpha}$. For
$\delta>0$, pick $N_{0}\in\mathbb{N}$ so that $2n^{-\alpha}<\delta$ for $n\geq
N_{0}$. Then, for any $N\geq N_{0}$

$\mathcal{H}^{s}_{\delta}(W)\leq\sum_{n=N}^{\infty}(2n^{-\alpha})^{s}=2^{-1-\alpha\epsilon}\sum_{n=N}^{\infty}n^{-1-\alpha\epsilon}.$

The latter tends to $0$ as $N$ tends to infinity, so for any $\delta>0$,
$\mathcal{H}^{s}_{\delta}(W)=0$, whence $\mathcal{H}^{s}(W)=0$. But then,
$\operatorname{dim_{H}}(W)\leq s=1/\alpha+\epsilon$. As $\epsilon>0$ was
arbitrary, we are done. ∎

###### Proof of Theorem 3.

By Lemma 11, we need only prove the lower bound. We let $\mu_{N}$ be the
sequence of probability measures defined by

$\frac{\mathrm{d}\mu_{N}}{\mathrm{d}x}=\frac{1}{N}\sum_{k=N+1}^{2N}\frac{1}{2(2N)^{-\alpha}}\mathbbm{1}_{B(a_{k}x,r_{N})},\qquad
r_{N}=(2N)^{-\alpha},$

and consider

$I_{t}(\mu_{N})=\iint|x-y|^{-t}\,\mathrm{d}\mu(x)\mathrm{d}\mu(y)=c_{0}\int|\hat{\mu}_{N}(\xi)|^{2}|\xi|^{t-1}\,\mathrm{d}\xi,$

where $c_{0}$ is a constant which does not depend on $N$ and $t$.

We have

$\frac{\mathrm{d}\mu_{N}}{\mathrm{d}x}=\frac{1}{N}\sum_{k=N+1}^{2N}\frac{1}{2(2N)^{-\alpha}}\mathbbm{1}_{B(0,r_{N})}*\delta_{a_{k}x},$

so that

$\hat{\mu}_{N}(\xi)=(2N)^{\alpha-1}\sum_{k=N+1}^{2N}\widehat{\mathbbm{1}_{B(0,r_{N})}}(\xi)e^{-i2\pi\xi
a_{k}x},$

and

$|\hat{\mu}_{N}(\xi)|^{2}=(2N)^{2\alpha-2}\sum_{k=N+1}^{2N}\sum_{l=N+1}^{2N}|\widehat{\mathbbm{1}_{B(0,r_{N})}}(\xi)|^{2}e^{-i2\pi\xi(a_{k}-a_{l})x}.$

Since

$\widehat{\mathbbm{1}_{B(0,r_{N})}}(\xi)=r_{N}\frac{\sin(2\pi r_{n}\xi)}{\pi
r_{N}\xi},$

we can estimate that

$|\widehat{\mathbbm{1}_{B(0,r_{N})}}(\xi)|^{2}\leq
m_{N}(\xi):=\min\\{2r_{N},\pi|\xi|^{-1}\\}^{2}.$

It follows that

$\displaystyle\int|\hat{\mu}_{N}(\xi)|^{2}\,\mathrm{d}\mu$
$\displaystyle\leq(2N)^{2\alpha-2}\sum_{k=N+1}^{2N}\sum_{l=N+1}^{2N}\int
m_{N}(\xi)e^{-i2\pi\xi(a_{k}-a_{l})x}\,\mathrm{d}\mu(x)$
$\displaystyle=(2N)^{2\alpha-2}m_{N}(\xi)\sum_{k=N+1}^{2N}\sum_{l=N+1}^{2N}\hat{\mu}((a_{k}-a_{l})\xi).$

Take constants $c$ and $\eta>0$ such that $|\hat{\mu}(\xi)|\leq
c\min\\{1,|\xi|^{-\eta}\\}$. Then, we have

$\sum_{k=N+1}^{2N}\sum_{l=N+1}^{2N}\hat{\mu}((a_{k}-a_{l})\xi)\leq cN^{2}.$

If $|\xi|\geq 1$, we can use that $(a_{k})$ is lacunary to prove that

$\sum_{k=N+1}^{2N}\sum_{l=N+1}^{2N}\hat{\mu}((a_{k}-a_{l})\xi)\leq\sum_{k=N+1}^{2N}\sum_{l=N+1}^{2N}c\min\\{1,|a_{k}-a_{l}|^{-\eta}\\}\leq
CN,$

for some constant $C$. We thus have

$\int|\hat{\mu}_{N}|^{2}\,\mathrm{d}\mu\leq c(2N)^{2\alpha}m_{N}(\xi),$

and if $|\xi|\geq 1$, we have

$\int|\hat{\mu}_{N}|^{2}\,\mathrm{d}\mu\leq C(2N)^{2\alpha-1}m_{N}(\xi).$

Combining the last two estimates, we get that

$\displaystyle\int I_{t}(\mu_{N})\,\mathrm{d}\mu$
$\displaystyle=c_{0}\iint|\hat{\mu}_{N}(\xi)|^{2}|\xi|^{t-1}\,\mathrm{d}\xi\,\mathrm{d}\mu$
$\displaystyle=c_{0}\iint|\hat{\mu}_{N}(\xi)|^{2}\,\mathrm{d}\mu\,|\xi|^{t-1}\,\mathrm{d}\xi$
$\displaystyle\leq
c^{\prime}N^{2\alpha}\int_{0}^{1}m_{N}(\xi)|\xi|^{t-1}\,\mathrm{d}\xi+C^{\prime}N^{2\alpha-1}\int_{1}^{\infty}m_{N}(\xi)|\xi|^{t-1}\,\mathrm{d}\xi.$

We can now easily compute that

$\displaystyle N^{2\alpha}\int_{0}^{1}m_{N}(\xi)|\xi|^{t-1}\,\mathrm{d}\xi$
$\displaystyle\leq
N^{2\alpha}r_{N}^{2}\int_{0}^{1}\xi^{t-1}\,\mathrm{d}\xi=c_{1},$
$\displaystyle
N^{2\alpha-1}\int_{1}^{\frac{\pi}{2}r_{N}^{-1}}m_{N}(\xi)|\xi|^{t-1}\,\mathrm{d}\xi$
$\displaystyle=4N^{2\alpha-1}r_{N}^{2}\int_{1}^{\frac{\pi}{2}r_{N}^{-1}}\xi^{t-1}\,\mathrm{d}\xi\leq
c_{2}N^{-1+\alpha t},$ $\displaystyle
N^{2\alpha-1}\int_{\frac{\pi}{2}r_{N}^{-1}}^{\infty}m_{N}(\xi)|\xi|^{t-1}\,\mathrm{d}\xi$
$\displaystyle=2\pi^{2}CN^{2\alpha-1}\int_{\frac{\pi}{2}r_{N}^{-1}}^{\infty}\xi^{t-3}\,\mathrm{d}\xi\leq
c_{3}N^{-1+\alpha t},$

where $c_{1}$, $c_{2}$ and $c_{3}$ do not depend on $N$. Hence

$\int I_{t}(\mu_{N})\,\mathrm{d}\mu\leq c_{4}+c_{5}N^{-1+\alpha t},$

which is uniformly bounded in $N$ provided $t<1/\alpha$.

Fix $t<1/\alpha$. It follows that for $\mu$-almost all $x$, there is a
subsequence $(N_{k})$ of the natural numbers, such that $I_{t}(\mu_{N_{k}})$
is uniformly bounded in $k$. Lemma 10 finishes the proof. ∎

###### Proof of Theorem 5.

As before, the upper bound is taken care of by Lemma 11. Once again, we define
probability measures $\mu_{N}$ with densities

$\frac{\mathrm{d}\mu_{N}}{\mathrm{d}x}=\frac{1}{N}\sum_{n=N+1}^{2N}\frac{1}{2(2N)^{-\alpha}}\mathbbm{1}_{B(x_{n},(2N)^{-\alpha})}.$

Then

$\operatorname{supp}\mu_{N}=E_{N}=\bigcup_{n=N+1}^{2N}B(x_{n},(2N)^{-\alpha}).$

Hence $E_{N}\subset W_{N}$, where

$W_{N}=\bigcup_{n=N+1}^{2N}B(x_{n},n^{-\alpha}).$

It follows that $\limsup E_{N}\subset W=\limsup W_{N}$.

Let $s=\eta/\alpha$. We show that there is a constant $c_{1}$ such that

$\mu_{N}(I)\leq c_{1}|I|^{s}$

holds for every $N$ and every interval $I$ (and hence for every Borel set).

Let $M(I,N)$ denote the number of $n$ with $N+1\leq n\leq 2N$ and $x_{n}\in
I$. We have

$M(N,I)\leq N|I|+\tilde{D}_{N}((x_{n}))\leq N|I|+cN^{1-\eta}.$

In every point $y\in[0,1)$, the number of $n=N+1,\ldots,2N$ such that $y\in
B(x_{n},(2N)^{-\alpha})$ is at most

$\sup_{|I|=2(2N)^{-\alpha}}M(I,N)\leq
2^{1-\alpha}N^{1-\alpha}+cN^{1-\eta}<(1+c)N^{1-\eta}.$

Hence, the density of $\mu_{N}$ satisfies

(9)
$\frac{\mathrm{d}\mu_{N}}{\mathrm{d}x}\leq\frac{1}{N}\frac{1}{2(2N)^{-\alpha}}(1+c)N^{1-\eta}\leq
2^{\alpha-1}(1+c)N^{\alpha-\eta}.$

If $|I|<N^{-\alpha}$, we use (9) to estimate that

$\displaystyle\mu_{N}(I)$ $\displaystyle\leq
2^{\alpha-1}(1+c)N^{\alpha-\eta}|I|$
$\displaystyle=2^{\alpha-1}(1+c)N^{\alpha-\eta}|I|^{1-\eta/\alpha}|I|^{\eta/\alpha}\leq
2^{\alpha-1}(1+c)|I|^{\eta/\alpha}.$

If $|I|\geq N^{-\alpha}$, let $3I$ denote the interval concentric with $I$ and
such that $|3I|=3|I|$. Then

(10) $\mu_{N}(I)\leq\frac{1}{N}M(3I,N)\leq 3|I|+cN^{-\eta}\leq
3|I|+c|I|^{\eta/\alpha}\leq(3+c)|I|^{\eta/\alpha}.$

This shows that $\mu_{N}(I)\leq c_{2}|I|^{s}$ with $c_{2}=2^{\alpha-1}(3+c)$.

It now follows for $t<s$ that

$\displaystyle I_{t}(\mu_{N})$
$\displaystyle=\iint|x-y|^{-t}\,\mathrm{d}\mu_{N}(x)\mathrm{d}\mu_{N}(y)$
$\displaystyle=\int\int_{0}^{\infty}\mu_{N}(B(x,u^{-1/t}))\,\mathrm{d}u\,\mathrm{d}\mu_{N}(x)$
$\displaystyle\leq\int\int_{0}^{\infty}\min\\{1,c_{2}(u^{-1/t})^{s}\\}\,\mathrm{d}u\,\mathrm{d}\mu_{N}(x)$
$\displaystyle=\int_{0}^{\infty}\min\\{1,c_{2}u^{-s/t}\\}\,\mathrm{d}u=:C<\infty.$

Since $D_{N}((x_{n}))\leq o(N)$, the measures $\mu_{N}$ converges weakly to
$\lambda$ and it follows from Lemma 10 that $\operatorname{dim_{H}}\limsup
E_{N}\geq t$. Since $t$ can be taken arbitrary close to $s=\eta/\alpha$ we
have $\operatorname{dim_{H}}W\geq\operatorname{dim_{H}}\limsup
E_{N}\geq\eta/\alpha$. ∎

###### Proof of Theorem 7.

The proof is similar to that of Theorem 5. We define $\mu_{N}$ in the same
way. It is then enough to prove that there is a constant $c_{3}$ such that
$\mu_{N}(I)\leq c_{3}|I|^{\frac{1}{\alpha}}$ holds for all $N$ and all
intervals $I$. We consider three cases.

First, we assume that $|I|\leq N^{-\beta}$. Then $I$ intersects at most two
balls $B(x_{n},(2N)^{-\alpha})$. Therefore, we have

$\mu_{N}(I)\leq|I|\frac{2}{N}\frac{1}{2(2N)^{-\alpha}}\leq
2^{\alpha}|I|N^{\alpha-1}\leq
2^{\alpha}|I|^{1-\frac{1}{\alpha}(\alpha-1)}=2^{\alpha}|I|^{\frac{1}{\alpha}}.$

If $\beta>\eta\alpha$, then we assume that $N^{-\beta}\leq|I|\leq
N^{-\eta\alpha}$. Otherwise, we do not have to consider this case.

The assumption $\beta\leq\eta(\alpha-1)+1$ implies that $\alpha\geq\beta$.
Hence, $I$ intersects a ball $B(x_{n},(2N)^{-\alpha})$ only if $x_{n}\in 3I$.
The number of such $x_{n}$ is at most $3|I|/(2N)^{-\beta}$ and

$\mu_{N}(I)\leq
2^{\beta}3|I|N^{\beta}\frac{1}{N}=2^{\beta}3|I|^{\frac{1}{\alpha}}|I|^{1-\frac{1}{\alpha}}N^{\beta-1}\leq
2^{\beta}3|I|^{\frac{1}{\alpha}}N^{-\eta(\alpha-1)+\beta-1}.$

Since $-\eta(\alpha-1)+\beta-1\leq 0$ by assumption, we have $\mu_{N}(I)\leq
2^{\beta}3|I|^{\frac{1}{\alpha}}$.

Finally, assume that $|I|\geq N^{-\eta\alpha}$. Then we have as in (10) that

$\mu_{N}(I)\leq 3|I|+cN^{-\eta}\leq(3+c)|I|^{\frac{1}{\alpha}}.$

All estimates taken together, we have for any interval $I$ and any $N$ that
$\mu_{N}(I)\leq c_{3}|I|^{\frac{1}{\alpha}}$, with
$c_{3}=\max\\{2^{\alpha},2^{\beta}3,3+c\\}$. The rest of the proof is exactly
as in the proof of Theorem 5. ∎

###### Proof of Corollary 8.

Let $\eta<\frac{1}{2}$. Then $D_{N}((q_{n}y))\leq cN^{1-\eta}$ for
$\lambda$-a.e. $y\in[0,1)$.

Let $\beta>2$ and consider the set

$\Delta=\\{\,|q_{l}-q_{k}|:k\neq l\,\\}.$

Let $(n_{j})_{j=1}^{\infty}$ be an enumeration of $\Delta$ such that if
$q_{l}-q_{k}\in\Delta$, then there is a $j\leq l^{2}$ such that
$n_{j}=q_{l}-q_{k}$. (Such an enumeration clearly exists.)

Consider the set

$V=\\{\,y\in[0,1):\lVert n_{j}y\rVert<j^{-\beta/2}\text{ for infinitely many
}n\in\mathbb{N}\,\\}.$

Since for each $j$,

$\lambda\\{\,y\in[0,1):\lVert
n_{j}y-\gamma\rVert<j^{-\beta/2}\,\\}=2j^{-\beta/2},$

we have by the first Borel–Cantelli lemma that $\lambda(V)=0$. It follows that
for $\lambda$-a.e. $y$ there exists $c>0$ such that $\lVert
n_{j}y\rVert>cj^{-\beta/2}$ for all $j$.

Suppose now that $y$ is such a typical point, and consider the sequence
$(q_{n}y)$. Take $1\leq k<l\leq N$ and let $j\leq l^{2}$ be such that
$n_{j}=q_{l}-q_{k}\in\Delta$. Since $j\leq N^{2}$, and by the choice of $y$,
we have

$\lVert q_{k}y-q_{l}y\rVert=\lVert(q_{l}-q_{k})y\rVert=\lVert
n_{j}y\rVert>cj^{-\beta/2}\geq cN^{-\beta}.$

This shows that $d_{N}((q_{n}y))>cN^{-\beta}$. It follows by Theorem 7 that
$\operatorname{dim_{H}}W_{y,\alpha}=\frac{1}{\alpha}$ whenever $\alpha$ is
such that $\beta\leq\eta(\alpha-1)+1$. Since $\beta$ can be taken as close to
$2$ and $\eta$ as close to $\frac{1}{2}$ as we desire,
$\operatorname{dim_{H}}W_{y,\alpha}=\frac{1}{\alpha}$ holds for almost all $y$
as long as $\alpha>3$. The case $\alpha=3$ follows by a limiting proceedure. ∎

##  5\. Concluding remarks

We strongly suspect that — as in the case of Lebesgue measure — the conclusion
of Theorem 1 can be strengthened from positive to full measure. However, the
‘inflation argument’ of Cassels used in the proof of Theorem 2 is not
immediately applicable unless both measures are Lebesgue or at least have the
same scaling properties. Hence, we do not at present know how to extend this.

In Corollary 8, the exact value of the Hausdorff dimension is calculated, but
only for $\alpha\geq 3$. Again, it would be natural to suspect that this can
be extended to $\alpha>1$, but our methods do not allow us to do this.

As a final remark, in the paper of Pollington, Velani, Zafeiropoulos and Zorin
[12], a weaker requirement on the Fourier decay of the measure $\mu$ is
required. It is possible that positive Fourier dimension can be weakened to,
say, logarithmic decay. We have not attempted to do this in the present
manuscript.

## References

  * [1] J. W. S. Cassels, _An introduction to Diophantine approximation_ , Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Hafner Publishing Co., New York, 1972, Facsimile reprint of the 1957 edition. MR 349591
  * [2] S. Chow and N. Technau, _Dispersion and littlewood’s conjecture_ , arXiv:2307.14871 (2023).
  * [3] S. Chow and A. Zafeiropoulos, _Fully inhomogeneous multiplicative Diophantine approximation of badly approximable numbers_ , Mathematika 67 (2021), no. 3, 639–646. MR 4258964
  * [4] K. L. Chung and P. Erdös, _On the application of the Borel-Cantelli lemma_ , Trans. Amer. Math. Soc. 72 (1952), 179–186. MR 45327
  * [5] K. J. Falconer, _Sets with large intersection properties_ , J. London Math. Soc. (2) 49 (1994), no. 2, 267–280. MR 1260112
  * [6] H. Furstenberg, _Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation_ , Math. Systems Theory 1 (1967), 1–49. MR 213508
  * [7] A. Haynes, J. L. Jensen, and S. Kristensen, _Metrical musings on Littlewood and friends_ , Proc. Amer. Math. Soc. 142 (2014), no. 2, 457–466. MR 3133988
  * [8] J.-P. Kahane and R. Salem, _Ensembles parfaits et séries trigonométriques_ , Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1301, Hermann, Paris, 1963. MR 160065
  * [9] A. Ya. Khinchin, _Continued fractions_ , University of Chicago Press, Chicago, Ill.-London, 1964. MR 161833
  * [10] P. Mattila, _Fourier analysis and Hausdorff dimension_ , Cambridge Studies in Advanced Mathematics, vol. 150, Cambridge University Press, Cambridge, 2015. MR 3617376
  * [11] T. Persson and H. W. J. Reeve, _A Frostman-type lemma for sets with large intersections, and an application to diophantine approximation_ , Proc. Edinb. Math. Soc. (2) 58 (2015), no. 2, 521–542. MR 3341452
  * [12] A. D. Pollington, S. Velani, A. Zafeiropoulos, and E. Zorin, _Inhomogeneous Diophantine approximation on $M_{0}$-sets with restricted denominators_, Int. Math. Res. Not. IMRN (2022), no. 11, 8571–8643. MR 4425845
  * [13] Z. Rudnick and P. Sarnak, _The pair correlation function of fractional parts of polynomials_ , Comm. Math. Phys. 194 (1998), no. 1, 61–70. MR 1628282
  * [14] N. Technau and A. Zafeiropoulos, _The discrepancy of $(n_{k}x)^{\infty}_{k=1}$ with respect to certain probability measures_, Q. J. Math. 71 (2020), no. 2, 573–597. MR 4112677