# Video ChatCaptioner:
Towards Enriched Spatiotemporal Descriptions

Jun Chen Deyao Zhu Kilichbek Haydarov Xiang Li Mohamed Elhoseiny
King Abdullah University of Science and Technology
{jun.chen, deyao.zhu, kilichbek.haydarov, xiang.li.1,
<EMAIL_ADDRESS>

###### Abstract

Video captioning aims to convey dynamic scenes from videos using natural
language, facilitating the understanding of spatiotemporal information within
our environment. Although there have been recent advances, generating detailed
and enriched video descriptions continues to be a substantial challenge. In
this work, we introduce _Video ChatCaptioner_ , an innovative approach for
creating more comprehensive spatiotemporal video descriptions. Our method
employs a ChatGPT model as a controller, specifically designed to select
frames for posing video content-driven questions. Subsequently, a robust
algorithm is utilized to answer these visual queries. This question-answer
framework effectively uncovers intricate video details and shows promise as a
method for enhancing video content. Following multiple conversational rounds,
ChatGPT can summarize enriched video content based on previous conversations.
We qualitatively demonstrate that our Video ChatCaptioner can generate
captions containing more visual details about the videos. The code is publicly
available at https://github.com/Vision-CAIR/ChatCaptioner

Figure 1: Video ChatCaptioner utilizes a ChatGPT to ask visual questions about
a specific video, without access to the visual content. BLIP-2 serves as the
answerer, providing answers to these questions based on the video content.
Following multiple rounds of this interactive question-and-answer
conversation, ChatGPT summarizes an enriched video description.

##  1 Introduction

Video captioning aims to automatically generate natural language descriptions
by analyzing the visual information present in a given video. This technology
offers numerous practical benefits, such as aiding the visually impaired [20],
facilitating robotic navigation [13], and enhancing autopilot systems [28].
Unlike image understanding, video captioning focuses on the spatiotemporal
analysis of a dense sequence of frames that typically contain a wealth of
visual information, including motion, people, and objects. Despite its
potential, accurately capturing and representing the rich visual details
within videos remains a formidable challenge for current prevailing
approaches.

Enriched video captioning offers a more comprehensive understanding of video
content for viewers. However, current methods face difficulties in generating
detailed video captions due to two primary limitations. First, ground-truth
captions in existing datasets tend to emphasize simple relationships or
activities within a video, frequently neglecting more intricate information,
such as more detailed descriptions of people, background elements, or specific
actions. Second, the scale of video-caption datasets is considerably smaller
compared to image-caption datasets. For example, the largest publicly
available image-caption dataset, LAION-5B [30], comprises 5 billion image-text
pairs, while the largest video-text datasets, such as HowTo100M [21] and
WebViD-10M [4], contain only 1.22 million and 10.7 million video-caption
pairs, respectively. This disparity in scale poses a challenge for AI models
to effectively learn and generate enriched diverse video descriptions.

To tackle these challenges and produce more comprehensive video descriptions,
we introduce Video ChatCaptioner. This method seeks to augment video
understanding through a dialogue between the ChatGPT [24] and BLIP-2 [17]
models. While BLIP-2 is pretrained on hundreds of millions of image-text
pairs, it struggles to process spatiotemporal data efficiently. The central
innovation of Video ChatCaptioner involves utilizing ChatGPT as a controller,
which then asks video content-driven questions to BLIP-2 for the sampled video
frames to obtain a spatiotemporal understanding of the entire video. Finally,
ChatGPT generates a video description based on the gathered question-answer
pairs. Therefore, This approach holds the promise to eliminate the need for
video-text datasets or models explicitly pretrained on such datasets, and also
offers the potential to generate more enriched video captions. Additionally,
we provide an illustrative example in Fig. 1, depicting an example of our
Video ChatCaptioner. It can be observed that Video ChatCaptioner produces more
detailed descriptions in comparison to the ground truth. For examples,
snowboarding tricks, snowy mountain, doing flips and balancing on a pole, and
helmet.

We evaluated the performance of Video ChatCaptioner by generating captions for
videos sampled from the MSVD [5] and WebVid [4] datasets. We conducted a
qualitative comparison between the generated captions and the ground truth.
Our observations revealed that Video ChatCaptioner is capable of producing
more comprehensive and informative video captions compared to ground-truth
captions. Additionally, we demonstrated that our designed frameworks can
generate a diverse set of visual questions, ensuring the extraction of more
video content. Overall, Video ChatCaptioner presents a new paradigm for video
caption generation, offering a valuable tool for crafting more intricate
descriptions that effectively convey spatio-temporal information.

##  2 Related work

Advancements in Image and Video Captioning. Recent developments in pretraining
models on extensive vision-language datasets have led to remarkable progress
in both image [7, 34, 14, 3, 6, 18, 17, 45, 25, 37] and video [42, 45, 4, 2,
10, 11, 15, 16, 19, 22, 31, 32, 35, 41] understanding. Significant
advancements have been made in the field of image and video captioning, as
demonstrated by various studies [6, 34, 3, 38, 18, 17, 40, 42, 45, 25]. Upon
pretraining on large-scale vision-language corpora, the resulting models are
capable of generating more diverse and accurate descriptions for images or
videos [17, 42, 25]. For instance, BLIP-2 [17] and Flamingo [3] exhibit
remarkable image captioning performance after being pretrained on hundreds of
millions of image-text pairs. Similarly, Vid2Seq [42] demonstrates improved
temporal event localization and description accuracy after pretraining on an
18-million YT-Temporal-1B video dataset [46]. In more recent works,
ChatCaptioner [47] explores the generation of more enriched image captions
through interactions between ChatGPT [24] and BLIP-2, where ChatGPT serves for
visual question asking, and BLIP-2 answers the questions. Our work is closely
aligned with ChatCaptioner, with a focus on facilitating conversations between
ChatGPT and BLIP-2 to produce richer video descriptions but more focusing on
the spatiotemporal temporal understanding.

Leveraging Pre-trained LLMs in Vision-Language Tasks. In recent years,
research on employing autoregressive language models as decoders in vision-
language tasks has gained significant momentum. This approach capitalizes on
cross-modal transfer, enabling the transfer of knowledge between language and
multimodal domains. VisualGPT [6] was among the first to demonstrate the data
efficiency of utilizing a pre-trained language model, specifically GPT-2 [27],
in image captioning tasks. Building on this success, Frozen [34] integrated a
pre-trained vision encoder with GPT-2, further enhancing its performance.
Subsequently, Flamingo [3] introduced a gated cross-attention dense block to
align the language model with a pre-trained vision encoder, while BLIP-2 [17]
proposed a Q-Former that transforms visual features into learnable visual
tokens, making them more accessible to language models. KOSMAS-1 [14] further
extended this approach by aligning various perception types (e.g., webpages,
mathematical equations, and visual explanations) with a pre-trained LLM.
Recently, after extensive pretraining, GPT-4 [25] has exhibited even more
robust and strong visual understanding capabilities, emphasizing the continued
advancements in this field.

Interaction between LLMs and Other Modules. Large language models have
demonstrated an impressive ability to improve their performance by integrating
with external modules [44, 29, 39, 47, 43, 33, 26]. For example, Toolformer
[29] fine-tunes a 6.7B GPT-J model [36] on a self-generated dataset with API-
usage examples, enabling it to use external tools such as Wikipedia search
API. ReAct [44] prompts LLMs to generate reason traces explicitly, guiding the
usage of external modules step-by-step for better performance. Visual ChatGPT
[39] and MM-ReAct [43] utilize ChatGPT’s in-context learning ability to teach
it how to use various vision models, enabling it to handle vision-related
tasks. Instead of using external modules with fixed API, ChatCaptioner [47]
employs another LLM model, BLIP-2 [17], as the external module, allowing
ChatGPT to interact with it through natural language. This conversation
between AI models results in better image captions with enriched details.
Camel [12] and DERA [23] have further demonstrated the potential of
conversational LLMs in programming and clinical tasks, respectively. However,
unlike ChatCaptioner, which is designed solely to describe images, Video
ChatCaptioner leverages the temporal understanding ability of ChatGPT,
empowering LLMs to generate rich and detailed descriptions of videos.

##  3 Method

Our Video ChatCaptioner is designed to capture the temporal relationships
among distinct spatial features by leveraging the automatic questioning and
summarization capabilities of ChatGPT. To achieve this, ChatGPT poses a
variety of questions to BLIP-2, a vision-language model proficient in
answering diverse visual queries. Although BLIP-2 is only trained on image-
text pairs, it can effectively infer various motion information, such as
dancing, sweeping, and riding, by only referencing individual frames. However,
a single BLIP-2 model cannot adequately represent the complete temporal
information within a video. Therefore, we propose Video ChatCaptioner, in Fig.
2, which treats ChatGPT as a controller to ask visual questions across
different frames and asks BLIP-2, as a VQA model, to provide the answers.
After multiple conversations between ChatGPT and BLIP-2, ChatGPT will
aggregate the spatiotemporal information together to summarize the video
content.

Figure 2:  Video ChatCaptioner Pipeline: Video ChatCaptioner incorporates two
primary components: a questioner, ChatGPT, responsible for posing questions,
and an answerer, BLIP-2, for answering the visual questions. With our designed
instructions, ChatGPT is prompted to select a frame from a set of uniformly
sampled frames and generate questions. Subsequently, BLIP-2 provides answers
to the visual questions based on the chosen video frame. Finally, ChatGPT
synthesizes a more enriched video caption from prior conversations.  Figure 3:
Prompt design: We showcase the prompt system designed for our Video
ChatCaptioner, which incorporates task instruction, question, answer and
summary prompts to facilitate effective communication between ChatGPT and
BLIP-2.

In order to achieve our goal, we initially uniform sample $N$ frames from a
given video, and we ask ChatGPT to ask informative questions for each frame.
Since ChatGPT does not have visual perception ability, we ask BLIP-2 to answer
the visual questions from ChatGPT. We have developed a prompt system to guide
ChatGPT in generating a diverse set of informative questions. We demonstrate
our designed prompt in Fig 3. Our prompt system comprises several levels,
including task instruction, sub-question instruction, and summary instruction,
each designed to enhance the quality of the questions generated. Additionally,
we have designed prompts to encourage BLIP-2 to generate more reliable
answers. In the following section, we will describe each level of the prompt
system in details.

###  3.1 Prompt Design

Task instruction for video understanding. The purpose of task instructions is
to offer a comprehensive context and outline the specific task for ChatGPT in
the domain of video content understanding. By doing so, it effectively steers
ChatGPT towards generating more targeted, video-content-related questions. The
effective communication between ChatGPT and BLIP-2 heavily depends on the
proper design of these instructions. They allow the information about the
video content to be enriched during each conversation round, leading to deeper
and more engaging interactions between the two agents.

Incorporating conversation logs. To enhance the quality of questions generated
by ChatGPT, we incorporate the conversation history prior to posing a new
question. This is achieved by pre-appending a series of prior question-answer
pairs to the question prompt. Supplying ChatGPT with this accumulated
contextual information can encourage the model to generate more insightful and
relevant questions.

Question prompt. The sub-question prompt aims to leverage the history of
question-answer pairs to generate more informative and relevant questions
while adhering to specific constraints. By reasoning from the available
question-answer pairs, the prompt provides explicit restrictions to guide the
generation of these questions, ensuring that they are coherent, relevant, and
aligned with the objectives. We demonstrate the sub-questions prompt as
follows.

For the first question, we use a fixed prompt to request a detailed
description from the first frame, as shown below:

Frame_1: Describe it in details.

For subsequent questions, we guide ChatGPT to formulate inquiries by defining
clear objectives, specifying the necessary actions, and establishing
constraints for the next question.

\- Frame sampling design: Rather than employing random frame sampling, we
encourage ChatGPT to devise a best frame sampling strategy on its own under
our designed instructions. This approach allows the model to select the frames
based on the current context to maximize its understanding of the whole video
content.

\- Question restriction: We impose specific limitations on the generation of
follow-up questions. (1) These questions must adhere to the format of
$Frame\\_ID$, enabling us to easily determine the frame ID with regular
expression for visual questioning. We also confine the frame selection range
to fall between 1 and N, the maximum sampled frame number, which can prevent
ChatGPT from selecting frames that are out-of-bounds. (2) Moreover, we
prohibit Video ChatCaptioner from posing questions related to unmentioned
individuals or objects from prior conversations. This measure aims to reduce
the occurrence of hallucinations during the visual question answering stage.
(3) Furthermore, to encourage the formulation of more informative inquiries,
we restrict ChatGPT from generating yes/no questions.

Figure 4: Illustrating ChatGPT and BLIP-2 conversation: We uniformly sample 9
frames from an MSVD video [5] and prompt ChatGPT to autonomously select frames
and generate 30 distinct questions for visual questioning

BLIP-2 answer prompt. We employ BLIP-2 to supply answers to ChatGPT’s
questions. As ChatGPT may pose highly open-ended and complex questions, BLIP-2
could potentially generate incorrect answers due to its language inductive
bias. To mitigate this issue, we have tailored the prompt to minimize such
occurrences. We allow BLIP-2 to respond with "do not know" if it is uncertain
about an answer, which has been empirically shown to reduce the number of
unconfident responses effectively.

Summary Prompt. In order to create a thorough and accurate summary of the
previous conversation between ChatGPT and BLIP2, we have carefully crafted a
prompt that encourages ChatGPT to synthesize the most relevant and precise
information from the prior question-answer exchanges. Examples of such prompts
can be found in Fig. 3.

##  4 Experiment

In this section, we will describe our Video ChatCaptioner from multiple
perspectives. First, we will discuss the details of experimental setting, the
primary experimental results and also ablation studies for different prompt
design.

Experimental Setup. Our experiment primarily involves ChatGPT [24] and BLIP-2
[17] engaging in conversation with each other to obtain spatiotemporal
perception of a given video. We specifically utilize the "gpt-3.5-turbo"
version of ChatGPT, accessible via the OpenAI API [1]. Additionally, we employ
the BLIP-2 version, which incorporates a FlanT5-XXL [8] language model and a
ViT-G vision encoder from the EVA-CLIP framework [9]. We sample the images
from WebViD [4] and MSVD [5] datasets.

Example Demonstration: To illustrate the interaction between ChatGPT and
BLIP-2, we randomly selected a sample from the MSVD dataset [5] and presented
the complete conversation along with the resulting summarized video
description in Fig. 4. We compare the generated description with the ground
truth, it demonstrates that our Video ChatCaptioner can provide a more
detailed description of the video’s content. For instance, it successfully
identifies key elements such as the small white chihuahua puppy, blue ball
indoors, white surface, and chases and plays with the ball.

Optimizing frame ID selection in ChatGPT. ChatGPT may occasionally fail to
generate the desired Frame_ID format, such as producing an out-of-bounds frame
index or not adhering to the required format. To ensure that ChatGPT generates
the appropriate frame ID format, we introduce two prompts in the questions
prompt:  (1) Video ChatCaptioner must ask questions from frame 1 to frame N.
(2) The questions format must be Frame_ID: question. Subsequently, we employ a
regular expression to extract the frame ID. A qualitative ablation study,
presented in Fig. 6, demonstrates the effectiveness of these prompts in
refining frame ID selection.

Reference:a man is rapidly firing bullets at a drum filled with water.

Frame-centric sampling: The video shows a man in a wooded area wearing a hat
and shooting a rifle at various targets, including a white target, a red
target, a water spout, and a barrel. He is seen standing, crouching, and
aiming with a serious expression on his face. The man is also shown wearing a
jacket in some frames. The video captures the motion and activity of the man
as he shoots the gun and moves around the forest.

Video-centric sampling: A man is shooting a pistol in a wooded area. He is
wearing a hat and shooting at a stationary object with multiple shots at once.
He is shooting rapidly with two hands and towards the ground.

Figure 5: frame-centric vs. video-centric sampling. The key objects and
actions captured are highlighted in red color. Figure 6: The effect of without
applying the frame selection prompt on the question instruction.

Frame sampling strategy ablation. There are two primary approaches guiding
ChatGPT’s question-asking process. The first referrs to as the frame-centric
approach, where ChatGPT asks questions from each frame individually and
generates a complete caption for each frame. Finally, ChatGPT synthesizes a
video summary based on these individual frame captions. The second, the video-
centric approach, requires ChatGPT to function within a broader video context.
ChatGPT autonomously selects the frame indices for asking questions and, after
a series of queries, generates a video summary based on the prior
conversation.

To evaluate the effectiveness of these distinct methodologies, we conducted an
ablation study comparing the frame-centric and video-centric strategies using
the MSVD videos [5]. A qualitative example from this analysis is illustrated
in Fig 5. The results reveal that the frame-centric sampling strategy tends to
have generate more inaccurate description. For instance, it incorrectly
identifies a single shooting target as multiple entities, such as a white
target, red target, water spout, and a barrel. In contrast, the video-centric
strategy accurately recognizes the object as a stationary target. This
discrepancy arises because the frame-centric sampling approach fails to
preserve the temporal consistency of objects within the video.

Reference: two boys playing video game.

Video ChatCaptioner: Two boys are sitting on a white couch in a living room
playing video games. They are wearing black hoodies and playing with an Xbox
360 controller connected to a TV. A laptop is visible on the table in front of
them. The boys seem to be enjoying the game.

Reference: Elderly businessman calling on phone on the open cafe near sea.

Video ChatCaptioner: A man in a suit and hat is standing on a dock looking at
his phone and talking on it. He appears to be checking the time on his watch
as well. In the background, a city can be seen.

Reference:Outside in the water a man is kayaking during on some heavy waves.

Video ChatCaptioner: A man is kayaking and rafting down a river in a forested
area. The river has blue water and rocky terrain. The man is wearing a
wetsuit, and he is paddling with a paddle.

Figure 7: More sampled examples with our Video ChatCaptioner. The key objects and actions are highlighted in red color.  Per Dialogue | All Questions
---|---
26.5/29 | 3326/4350
91% | 76%
Table 1: Number of unique questions per dialogue and per all questions.

Question Diversity: In order to assess whether Video ChatCaptioner generates
relevant and diverse visual questions instead of repetitive ones, we evaluate
the uniqueness of the questions posed during dialogues and across the entire
dataset. We randomly select 150 images from the MSVD dataset and utilize the
Video ChatCaptioner to generate 30 questions for each conversation. Given that
the initial question is always hard-coded, we exclude it from our analysis.
The results, presented in Table 1, reveal that 91% of the questions within
each dialogue and 76% of the questions across the entire dataset are unique.
This substantial proportion of unique questions underscores the diversity of
the questions generated by Video ChatCaptioner.

Additional Qualitative Examples: In Fig 7, we present further qualitative
instances that showcase the superior performance of Video ChatCaptioner in
generating detailed and informative video description compared to the ground
truth. These enhanced descriptions prove to be more valuable in conveying the
nuances of the video content. For instance, in the first example, Video
ChatCaptioner accurately identifies that two girls are performing a
synchronized choreographed routine, dressed in pink and different short
dresses and dancing barefoot. These intricate details offer a more vivid and
comprehensive description of the video, which is absent in the ground truth
captions.

##  5 Limitation

Through the success of Video ChatCaptioner capable of generating more diverse
and detailed video descrition, the video ChatCaptioner system still suffers
from several limitations.

Limitations of the perception model: While BLIP-2 exhibits robust visual
perception and VQA capabilities, it occasionally misidentifies objects,
attributes, or visual relationships, which can compromise the accuracy of the
generated video captions. Moreover, as BLIP-2 is not specifically trained on
videos, it may struggle to capture intricate motion information. In addition,
due to its training on a static large language model, BLIP-2 inadvertently
inherits language biases present in the LLM, potentially generating content
that is not visually grounded. Addressing these limitations can be
accomplished by employing a more advanced perception model, such as GPT-4
[25].

Inference Speed: The Video ChatCaptioner system necessitates the collaboration
of a language model and a VQA (Visual Question Answering) model to generate
descriptive video captions through interactive conversations. As a result of
these multi-round conversations, the inference time for the Video
ChatCaptioner is typically longer compared to traditional video captioning
models. To optimize the inference speed, one possible solution is to instruct
the language model to formulate more informative questions so that the number
of conversation rounds can be minimized.

Temporal Identification of multiple people or objects: In Video ChatCaptioner,
ChatGPT remains blind to the actual video content, as it is only prompted to
ask visual questions and summarize the information from the video.
Consequently, it lacks an understanding of temporal consistency information.
For instance, Fig. 8 illustrates that a video depicts a group of people
lifting barbells in a gym, appearing in different frames. However, the Video
ChatCaptioner only mentions a man and a woman. This issue may arise from two
primary factors: 1) The uniform sampling of video frames fails to capture
objects in videos with rapidly changing frame rates, which hinders BLIP-2’s
ability to effectively process multiple objects or people information; 2)
ChatGPT can only identify objects or people from text conversations, lacking
visual grounding capabilities, which may result in insufficient coverage of
multiple objects or people in its output.

Reference: A large group of people are seen sitting around a gym with many
seen on the side lifting weights over their heads.

Ours: In the video, a man is seen lifting a barbell and doing squats in a gym.
The man is wearing a shirt and shorts, and is not using any other equipment
besides the barbell. A woman is also seen lifting weights in the background.
The gym also has a bench.

Figure 8: Illustrated example that fails to temporally capture the multiple
people

##  6 Discussion

In this work, we introduce a novel approach, named Video ChatCaptioner, which
generates enriched spatiotemporal descriptions through interactive
conversations between ChatGPT and BLIP-2. ChatGPT serves as a controller,
selecting frames for posing visual questions, while BLIP-2 provides answers to
these questions. Subsequently, ChatGPT synthesizes a comprehensive video
description by integrating information from the preceding dialogue. Through
many qualitative examples, we can observe that our method can generate richer
video descriptions compared to the ground-truth captions. Overall, our
approach provides a new paradigm for the video caption generation and holds
the potential to inspire more further research in this area.

## References

  * [1] Openai api: Gpt-3.5.  https://platform.openai.com/docs/models/gpt-3-5, 2023.  Accessed: March 7, 2023.
  * [2] Hassan Akbari, Liangzhe Yuan, Rui Qian, Wei-Hong Chuang, Shih-Fu Chang, Yin Cui, and Boqing Gong.  Vatt: Transformers for multimodal self-supervised learning from raw video, audio and text.  Advances in Neural Information Processing Systems, 34:24206–24221, 2021.
  * [3] Jean-Baptiste Alayrac, Jeff Donahue, Pauline Luc, Antoine Miech, Iain Barr, Yana Hasson, Karel Lenc, Arthur Mensch, Katherine Millican, Malcolm Reynolds, et al.  Flamingo: a visual language model for few-shot learning.  In Advances in Neural Information Processing Systems, 2022.
  * [4] Max Bain, Arsha Nagrani, Gül Varol, and Andrew Zisserman.  Frozen in time: A joint video and image encoder for end-to-end retrieval.  In IEEE International Conference on Computer Vision, 2021.
  * [5] David Chen and William B Dolan.  Collecting highly parallel data for paraphrase evaluation.  In Proceedings of the 49th annual meeting of the association for computational linguistics: human language technologies, pages 190–200, 2011\.
  * [6] Jun Chen, Han Guo, Kai Yi, Boyang Li, and Mohamed Elhoseiny.  Visualgpt: Data-efficient adaptation of pretrained language models for image captioning.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 18030–18040, 2022.
  * [7] Yen-Chun Chen, Linjie Li, Licheng Yu, Ahmed El Kholy, Faisal Ahmed, Zhe Gan, Yu Cheng, and Jingjing Liu.  Uniter: Universal image-text representation learning.  In Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XXX, pages 104–120. Springer, 2020.
  * [8] Hyung Won Chung, Le Hou, Shayne Longpre, Barret Zoph, Yi Tay, William Fedus, Eric Li, Xuezhi Wang, Mostafa Dehghani, Siddhartha Brahma, et al.  Scaling instruction-finetuned language models.  arXiv preprint arXiv:2210.11416, 2022.
  * [9] Yuxin Fang, Wen Wang, Binhui Xie, Quan Sun, Ledell Wu, Xinggang Wang, Tiejun Huang, Xinlong Wang, and Yue Cao.  Eva: Exploring the limits of masked visual representation learning at scale.  arXiv preprint arXiv:2211.07636, 2022.
  * [10] Tsu-Jui Fu, Linjie Li, Zhe Gan, Kevin Lin, William Yang Wang, Lijuan Wang, and Zicheng Liu.  Violet: End-to-end video-language transformers with masked visual-token modeling.  arXiv preprint arXiv:2111.12681, 2021.
  * [11] Yuying Ge, Yixiao Ge, Xihui Liu, Dian Li, Ying Shan, Xiaohu Qie, and Ping Luo.  Bridging video-text retrieval with multiple choice questions.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 16167–16176, 2022.
  * [12] Li Guohao, Hammoud Hasan Abed Al Kader, Itani Hani, Khizbullin Dmitrii, and Ghanem Bernard.  Camel: Communicative agents for “mind” exploration of large scale language model society, 2023.
  * [13] Chenguang Huang, Oier Mees, Andy Zeng, and Wolfram Burgard.  Visual language maps for robot navigation.  arXiv preprint arXiv:2210.05714, 2022.
  * [14] Shaohan Huang, Li Dong, Wenhui Wang, Yaru Hao, Saksham Singhal, Shuming Ma, Tengchao Lv, Lei Cui, Owais Khan Mohammed, Qiang Liu, et al.  Language is not all you need: Aligning perception with language models.  arXiv preprint arXiv:2302.14045, 2023.
  * [15] Jie Lei, Linjie Li, Luowei Zhou, Zhe Gan, Tamara L Berg, Mohit Bansal, and Jingjing Liu.  Less is more: Clipbert for video-and-language learning via sparse sampling.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 7331–7341, 2021.
  * [16] Dongxu Li, Junnan Li, Hongdong Li, Juan Carlos Niebles, and Steven CH Hoi.  Align and prompt: Video-and-language pre-training with entity prompts.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4953–4963, 2022.
  * [17] Junnan Li, Dongxu Li, Silvio Savarese, and Steven Hoi.  Blip-2: Bootstrapping language-image pre-training with frozen image encoders and large language models.  arXiv preprint arXiv:2301.12597, 2023.
  * [18] Junnan Li, Dongxu Li, Caiming Xiong, and Steven Hoi.  Blip: Bootstrapping language-image pre-training for unified vision-language understanding and generation.  In International Conference on Machine Learning, pages 12888–12900. PMLR, 2022.
  * [19] Linjie Li, Yen-Chun Chen, Yu Cheng, Zhe Gan, Licheng Yu, and Jingjing Liu.  Hero: Hierarchical encoder for video+ language omni-representation pre-training.  In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 2046–2065, 2020.
  * [20] Xingyu Liu, Patrick Carrington, Xiang’Anthony’ Chen, and Amy Pavel.  What makes videos accessible to blind and visually impaired people?  In Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems, pages 1–14, 2021.
  * [21] Antoine Miech, Dimitri Zhukov, Jean-Baptiste Alayrac, Makarand Tapaswi, Ivan Laptev, and Josef Sivic.  HowTo100M: Learning a Text-Video Embedding by Watching Hundred Million Narrated Video Clips.  In ICCV, 2019.
  * [22] Antoine Miech, Dimitri Zhukov, Jean-Baptiste Alayrac, Makarand Tapaswi, Ivan Laptev, and Josef Sivic.  Howto100m: Learning a text-video embedding by watching hundred million narrated video clips.  In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 2630–2640, 2019.
  * [23] Varun Nair, Elliot Schumacher, Geoffrey Tso, and Anitha Kannan.  Dera: Enhancing large language model completions with dialog-enabled resolving agents, 2023.
  * [24] OpenAI.  Introducing chatgpt.  https://openai.com/blog/chatgpt, 2022.
  * [25] OpenAI.  Gpt-4 technical report, 2023.
  * [26] Baolin Peng, Michel Galley, Pengcheng He, Hao Cheng, Yujia Xie, Yu Hu, Qiuyuan Huang, Lars Liden, Zhou Yu, Weizhu Chen, et al.  Check your facts and try again: Improving large language models with external knowledge and automated feedback.  arXiv preprint arXiv:2302.12813, 2023.
  * [27] Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al.  Language models are unsupervised multitask learners.  OpenAI blog, 1(8):9, 2019.
  * [28] Sharmin Rahman, Alberto Quattrini Li, and Ioannis Rekleitis.  Svin2: An underwater slam system using sonar, visual, inertial, and depth sensor.  In 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 1861–1868. IEEE, 2019.
  * [29] Timo Schick, Jane Dwivedi-Yu, Roberto Dessì, Roberta Raileanu, Maria Lomeli, Luke Zettlemoyer, Nicola Cancedda, and Thomas Scialom.  Toolformer: Language models can teach themselves to use tools.  arXiv preprint arXiv:2302.04761, 2023.
  * [30] Christoph Schuhmann, Romain Beaumont, Richard Vencu, Cade W Gordon, Ross Wightman, Mehdi Cherti, Theo Coombes, Aarush Katta, Clayton Mullis, Mitchell Wortsman, Patrick Schramowski, Srivatsa R Kundurthy, Katherine Crowson, Ludwig Schmidt, Robert Kaczmarczyk, and Jenia Jitsev.  LAION-5b: An open large-scale dataset for training next generation image-text models.  In Thirty-sixth Conference on Neural Information Processing Systems Datasets and Benchmarks Track, 2022.
  * [31] Paul Hongsuck Seo, Arsha Nagrani, Anurag Arnab, and Cordelia Schmid.  End-to-end generative pretraining for multimodal video captioning.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 17959–17968, 2022.
  * [32] Chen Sun, Austin Myers, Carl Vondrick, Kevin Murphy, and Cordelia Schmid.  Videobert: A joint model for video and language representation learning.  In Proceedings of the IEEE/CVF international conference on computer vision, pages 7464–7473, 2019.
  * [33] Dídac Surís, Sachit Menon, and Carl Vondrick.  Vipergpt: Visual inference via python execution for reasoning.  arXiv preprint arXiv:2303.08128, 2023.
  * [34] Maria Tsimpoukelli, Jacob L Menick, Serkan Cabi, SM Eslami, Oriol Vinyals, and Felix Hill.  Multimodal few-shot learning with frozen language models.  Advances in Neural Information Processing Systems, 34:200–212, 2021\.
  * [35] Alex Jinpeng Wang, Yixiao Ge, Rui Yan, Yuying Ge, Xudong Lin, Guanyu Cai, Jianping Wu, Ying Shan, Xiaohu Qie, and Mike Zheng Shou.  All in one: Exploring unified video-language pre-training.  arXiv preprint arXiv:2203.07303, 2022.
  * [36] Ben Wang and Aran Komatsuzaki.  GPT-J-6B: A 6 Billion Parameter Autoregressive Language Model.  https://github.com/kingoflolz/mesh-transformer-jax, May 2021.
  * [37] Wenhui Wang, Hangbo Bao, Li Dong, Johan Bjorck, Zhiliang Peng, Qiang Liu, Kriti Aggarwal, Owais Khan Mohammed, Saksham Singhal, Subhojit Som, et al.  Image as a foreign language: Beit pretraining for all vision and vision-language tasks.  arXiv preprint arXiv:2208.10442, 2022.
  * [38] Wenhui Wang, Hangbo Bao, Li Dong, Johan Bjorck, Zhiliang Peng, Qiang Liu, Kriti Aggarwal, Owais Khan Mohammed, Saksham Singhal, Subhojit Som, et al.  Image as a foreign language: Beit pretraining for all vision and vision-language tasks.  arXiv preprint arXiv:2208.10442, 2022.
  * [39] Chenfei Wu, Shengming Yin, Weizhen Qi, Xiaodong Wang, Zecheng Tang, and Nan Duan.  Visual chatgpt: Talking, drawing and editing with visual foundation models.  arXiv preprint arXiv:2303.04671, 2023.
  * [40] Haiyang Xu, Qinghao Ye, Ming Yan, Yaya Shi, Jiabo Ye, Yuanhong Xu, Chenliang Li, Bin Bi, Qi Qian, Wei Wang, et al.  mplug-2: A modularized multi-modal foundation model across text, image and video.  arXiv preprint arXiv:2302.00402, 2023.
  * [41] Hongwei Xue, Tiankai Hang, Yanhong Zeng, Yuchong Sun, Bei Liu, Huan Yang, Jianlong Fu, and Baining Guo.  Advancing high-resolution video-language representation with large-scale video transcriptions.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 5036–5045, 2022.
  * [42] Antoine Yang, Arsha Nagrani, Paul Hongsuck Seo, Antoine Miech, Jordi Pont-Tuset, Ivan Laptev, Josef Sivic, and Cordelia Schmid.  Vid2seq: Large-scale pretraining of a visual language model for dense video captioning.  arXiv preprint arXiv:2302.14115, 2023.
  * [43] Zhengyuan Yang, Linjie Li, Jianfeng Wang, Kevin Lin, Ehsan Azarnasab, Faisal Ahmed, Zicheng Liu, Ce Liu, Michael Zeng, and Lijuan Wang.  Mm-react: Prompting chatgpt for multimodal reasoning and action.  arXiv preprint arXiv:2303.11381, 2023.
  * [44] Shunyu Yao, Jeffrey Zhao, Dian Yu, Nan Du, Izhak Shafran, Karthik Narasimhan, and Yuan Cao.  React: Synergizing reasoning and acting in language models.  arXiv preprint arXiv:2210.03629, 2022.
  * [45] Lu Yuan, Dongdong Chen, Yi-Ling Chen, Noel Codella, Xiyang Dai, Jianfeng Gao, Houdong Hu, Xuedong Huang, Boxin Li, Chunyuan Li, et al.  Florence: A new foundation model for computer vision.  arXiv preprint arXiv:2111.11432, 2021.
  * [46] Rowan Zellers, Jiasen Lu, Ximing Lu, Youngjae Yu, Yanpeng Zhao, Mohammadreza Salehi, Aditya Kusupati, Jack Hessel, Ali Farhadi, and Yejin Choi.  Merlot reserve: Neural script knowledge through vision and language and sound.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 16375–16387, 2022.
  * [47] Deyao Zhu, Jun Chen, Kilichbek Haydarov, Xiaoqian Shen, Wenxuan Zhang, and Mohamed Elhoseiny.  Chatgpt asks, blip-2 answers: Automatic questioning towards enriched visual descriptions.  arXiv preprint arXiv:2303.06594, 2023.


# Extremal sequences for a weighted zero-sum constant

Santanu Mondal  Ramakrishna Mission Vivekananda Educational and Research
Institute, Belur, Dist. Howrah, 711202, India Krishnendu Paul  Ramakrishna
Mission Vivekananda Educational and Research Institute, Belur, Dist. Howrah,
711202, India Shameek Paul E-mail addresses<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Ramakrishna Mission
Vivekananda Educational and Research Institute, Belur, Dist. Howrah, 711202,
India

###### Abstract

For an abelian group $G$, the constant $C(G)$ is defined to be the smallest
natural number $k$, such that any sequence of $k$ elements in $G$ has a
subsequence of consecutive terms whose sum is zero (the identity element). We
characterize the extremal sequences for a weighted version of this constant
for some particular weights for a finite cyclic group.

Keywords: Extremal sequences, Zero-sum constants

##  1 Introduction

We denote the additive identity in an abelian group $(G,+)$ by 0.

######  Definition 1.

Let $(G,+)$ be a finite abelian group of exponent $m\geq 2$ and let
$A\subseteq\\{1,2,\ldots,m-1\\}$. For any $k\geq 1$, a sequence
$S:(x_{1},\ldots,x_{k})$ in $G$ is called an $A$-weighted zero-sum sequence if
for $1\leq i\leq k$, $\exists~{}a_{i}\in A$ such that
$a_{1}x_{1}+\ldots+a_{k}x_{k}=0$.

When $A=\\{1\\}$, an $A$-weighted zero-sum sequence is also called a zero-sum
sequence.

######  Definition 2.

For a finite abelian group $(G,+)$ of exponent $m\geq 2$ and for a subset
$A\subseteq\\{1,2,\ldots,m-1\\}$, the $A$-weighted Davenport constant of $G$,
denoted by $D_{A}(G)$, is defined to be the least positive integer $k$ such
that any sequence in $G$ of length $k$ has an $A$-weighted zero-sum
subsequence.

This constant was introduced in [1]. The following related constant was
introduced in [5] in Definition 5.

######  Definition 3.

For a finite abelian group $(G,+)$ of exponent $m\geq 2$ and for a subset
$A\subseteq\\{1,2,\ldots,m-1\\}$, we define the constant $C_{A}(G)$ to be the
least positive integer $k$ such that any sequence in $G$ of length $k$ has an
$A$-weighted zero-sum subsequence of consecutive terms.

For any $A$, we have $C_{A}(G)\geq D_{A}(G)$. When $A=\\{1\\}$, we denote
$C_{A}(G)$ by $C(G)$. For $n\geq 2$, let $C_{A}(n)$ denote
$C_{A}(\mathbb{Z}_{n})$.

In [2], extremal sequences were studied for some weight sets $A$ for the
$A$-weighted Davenport constant. In this paper, we study the extremal
sequences for this different weighted zero-sum constant. We begin with the
following definition.

######  Definition 4.

Let $A$ be a subset of $\mathbb{Z}_{n}\setminus\\{0\\}$ and let $k=C_{A}(n)$.
Then, by the definition of $C_{A}(n)$, there exists a sequence $S$ in
$\mathbb{Z}_{n}$ of length $k-1$ which has no $A$-weighted zero-sum
subsequence of consecutive terms. We will call such a sequence an $A$-extremal
sequence.

Remark: If we permute the terms of an $A$-extremal sequence, then we may not
get an $A$-extremal sequence. For example, when $A=\\{1,-1\\}$ and $S:(2,1,2)$
is a sequence in $\mathbb{Z}_{4}$, then $S$ does not have any $A$-weighted
zero-sum subsequence of consecutive terms. Also, from Corollary 3 of [5], we
have $C_{A}(4)=4$ and so $S$ is an $A$-extremal sequence. Clearly, $(1,2,2)$
is not an $A$-extremal sequence in $\mathbb{Z}_{4}$.

In Theorem 2 of [5], it has been shown that $C(n)=n$. The next theorem
characterizes the extremal sequences in $\mathbb{Z}_{n}$, when $A=\\{1\\}$.

######  Theorem 1.

Let $A=\\{1\\}$ and let $S:(x_{1},\ldots,x_{n-1})$ be a sequence in
$\mathbb{Z}_{n}$. Then, $S$ is an $A$-extremal sequence if and only if for
$1\leq i\leq n-1$, we have
$x_{i}\notin\\{0,-x_{i-1},\,-x_{i-1}-x_{i-2},\,\ldots\ldots,\,-x_{i-1}-x_{i-2}-\ldots-
x_{1}\\}$. There are $(n-1)\,!$ $A$-extremal sequences in $\mathbb{Z}_{n}$.

###### Proof.

Let $S:(x_{1},\ldots,x_{n-1})$ be a sequence in $\mathbb{Z}_{n}$. As $C(n)=n$,
so $S$ is an $A$-extremal sequence if and only if $S$ does not have any zero-
sum subsequence of consecutive terms. We now observe that for $1\leq i\leq
n-1$, we have
$x_{i}\in\\{0,-x_{i-1},\,-x_{i-1}-x_{i-2},\,\ldots\ldots,\,-x_{i-1}-x_{i-2}-\cdots-
x_{1}\\}$ if and only if $\exists~{}j\in\\{1,\ldots,i\\}$ such that
$x_{j}+x_{j+1}\ldots+x_{i-1}+x_{i}=0$.

To construct such a sequence, we can choose $x_{1}$ in $n-1$ ways. Given
$x_{1}$, we can choose $x_{2}$ in $n-2$ ways. Given $x_{1},x_{2}$, we can
choose $x_{3}$ in $n-3$ ways. We continue in this manner until, finally, given
$x_{1},x_{2},\ldots,x_{n-2}$, we have a unique choice for $x_{n-1}$. So, we
get $(n-1)\,!$ $A$-extremal sequences in $\mathbb{Z}_{n}$. ∎

In Theorem 3 of [5], it has been shown that, for
$A=\mathbb{Z}_{n}\setminus\\{0\\}$, we have $C_{A}(n)=2$.

######  Theorem 2.

For $A=\mathbb{Z}_{n}\setminus\\{0\\}$, a sequence $(x)$ of length 1, is an
$A$-extremal sequence if and only if $x$ is a unit.

###### Proof.

We observe that a sequence $(x)$ of length 1, is an $A$-weighted zero-sum
sequence if and only if $x$ is a zero-divisor. ∎

Let $U(n)$ denote the multiplicative group of units in the ring
$\mathbb{Z}_{n}$. For $j\geq 1$, let $U(n)^{j}$ denote the set $\\{\,x^{j}\mid
x\in U(n)\,\\}$. If $n=p_{1}p_{2}\ldots p_{k}$, where $p_{i}$ is a prime for
$1\leq i\leq k$, then, we define $\Omega(n)=k$.

Remark: Theorem 2 characterizes the $U(p)$-extremal sequences in
$\mathbb{Z}_{p}$, where $p$ is a prime.

In this article, the following are among some of the results which we have
obtained:

  * •

For $A=U(p),~{}U(p)^{2}$ and $U(p)^{3}$, where $p$ is a prime, we characterize
the $A$-extremal sequences in $\mathbb{Z}_{p}$.

  * •

For $A=U(n)$, where $n$ is odd, we characterize the $A$-extremal sequences in
$\mathbb{Z}_{n}$.

  * •

For $A=U(n)^{2}$, where every prime divisor of $n$ is at least 7, we
characterize the $A$-extremal sequences in $\mathbb{Z}_{n}$.

  * •

For $A=U(n)^{3}$, where $n$ is squarefree and is not divisible by 2, 7 and 13,
we characterize the $A$-extremal sequences in $\mathbb{Z}_{n}$.

##  2 The case when $A=U(p)^{2}$ or $A=U(p)^{3}$, where $p$ is a prime

When $p=2$, for any $j$, we have $U(2)^{j}=U(2)$. So, Theorem 2 characterizes
the $U(2)^{j}$-extremal sequences in $\mathbb{Z}_{2}$. In fact, $S:(1)$ is the
only $U(2)^{j}$-extremal sequence in $\mathbb{Z}_{2}$.

###  2.1 $A=U(p)^{2}$, where $p$ is an odd prime

For an odd prime $p$, we denote $U(p)^{2}$ by $Q_{p}$.

######  Definition 5.

Let $R$ be a ring and let $A$ be a subset of $R$. A sequence
$S:(x_{1},\ldots,x_{k})$ in $R$ is an $A$-weighted zero-sum sequence if
$\exists~{}a_{1},\ldots,a_{k}\in A$ such that
$a_{1}x_{1}+\ldots+a_{k}x_{k}=0$.

######  Theorem 3.

Let $F$ be a field and let $A$ be a subgroup of $F^{*}$. A sequence $S:(x,y)$
in $F^{*}$ has an $A$-weighted zero-sum subsequence $\iff S$ is an
$A$-weighted zero-sum sequence $\iff x$ and $-y$ are in the same coset of $A$.

###### Proof.

Let $S:(x,y)$ be a sequence in $F^{*}$. We now observe that $x$ and $-y$ are
in the same coset of $A\iff\exists~{}c\in A$ such that
$x=-cy\iff\exists~{}c\in A$ such that $x+cy=0\iff\exists~{}a,b\in A$ such that
$ax+by=0$. ∎

######  Corollary 1.

Let $F$ be a field and let $A$ be a subgroup of $F^{*}$. A sequence $S:(x,y)$
in $F$ does not have an $A$-weighted zero-sum subsequence if and only if
$x,y\in F^{*}$ and $x,-y$ are in different cosets of $A$ in $F^{*}$.

In Theorem 5 of [5], it has been shown that, when $A=Q_{p}$, where $p$ is a
prime, we have $C_{A}(2)=2$ and $C_{A}(p)=3$ when $p\neq 2$.

######  Corollary 2.

Let $A=Q_{p}$, where $p$ is an odd prime. Let $S:(x,y)$ be a sequence in
$\mathbb{Z}_{p}$. Then, $S$ is an $A$-extremal sequence in $\mathbb{Z}_{p}$ if
and only if $x,y\in U(p)$ and $x,-y$ are in different cosets of $A$ in $U(p)$.

###### Proof.

This follows from Corollary 1 by considering the field $F=\mathbb{Z}_{p}$ and
the subgroup $Q_{p}$ of $F^{*}$. ∎

Remark: Theorem 2 and Corollary 2 together characterize the
$U(p)^{2}$-extremal sequences in $\mathbb{Z}_{p}$, when $p$ is a prime.

###  2.2 $A=U(p)^{3}$, where $p$ is an odd prime

Let $p$ be a prime. The following three facts have been shown in [5] in
Theorem 6 and Lemma 1. When $p\not\equiv 1~{}(mod~{}3)$, $U(p)^{3}=U(p)$. When
$p\equiv 1~{}(mod~{}3)$, we have $C_{A}(p)=3$, except when $p\neq 7$. Also,
$C_{A}(7)=4$.

When $p\not\equiv 1~{}(mod~{}3)$, $A=\mathbb{Z}_{p}\setminus\\{0\\}$ and so
Theorem 2 characterizes the $A$-extremal sequences in $\mathbb{Z}_{p}$. The
next two results characterize the $A$-extremal sequences in $\mathbb{Z}_{p}$,
when $p\equiv 1~{}(mod~{}3)$.

######  Corollary 3.

Let $A=U(p)^{3}$, where $p$ is a prime such that $p\equiv 1~{}(mod~{}3)$ and
$p\neq 7$. Let $S:(x,y)$ be a sequence in $\mathbb{Z}_{p}$. Then, $S$ is an
$A$-extremal sequence in $\mathbb{Z}_{p}$ if and only if $x,y\in U(p)$ and
$x,-y$ are in different cosets of $A$ in $U(p)$.

###### Proof.

Let $S:(x,y)$ be a sequence in $\mathbb{Z}_{p}$. As $C_{A}(p)=3$, so $S$ is an
$A$-extremal sequence in $\mathbb{Z}_{p}$ if and only if $S$ has no
$A$-weighted zero-sum subsequence. Now, from Corollary 1, on considering
$\mathbb{Z}_{p}$ as a field, the result follows. ∎

The following definition is similar to a definition which has been made in
[2].

######  Definition 6.

Let $S:(x_{1},\ldots,x_{k})$ and $T:(y_{1},\ldots,y_{k})$ be two sequences in
$\mathbb{Z}_{n}$ and let $A$ be a subgroup of $U(n)$. We say that $S$ is
$A$-equivalent to $T$ if $\exists~{}c\in U(n)$ and for $1\leq i\leq k$,
$\exists~{}a_{i}\in A$ such that $y_{i}=c\,a_{i}x_{i}$.

Observe that if $S$ is an $A$-weighted zero-sum sequence, then, so is any
sequence which is $A$-equivalent to $S$. Thus, if $S$ is an $A$-extremal
sequence, then, so is any sequence which is $A$-equivalent to $S$.

######  Observation 1.

Let $A=U(7)^{3}$. Then, $S$ is an $A$-extremal sequence in $\mathbb{Z}_{7}$ if
and only if $S$ is $A$-equivalent to $(1,3,1)$.

###### Proof.

Let $S$ be a sequence in $\mathbb{Z}_{7}$. Suppose $S$ is an $A$-extremal
sequence. As $C_{A}(7)=4$, so $S$ has length 3. As $S$ has no $A$-weighted
zero-sum subsequence of consecutive terms, so all terms of $S$ are non-zero.
Thus, the terms of $S$ are in $\\{\pm 1,\pm 2,\pm 3\\}$. As $A=\\{1,-1\\}$, no
two consecutive terms of $S$ can be equal upto sign and $S$ cannot be
$A$-equivalent to $(1,2,1)$ or to any permutation of $(1,2,3)$. By multiplying
by a unit, we see that $S$ is $A$-equivalent to a sequence whose first term is
1. Thus, $S$ has to be $A$-equivalent to a sequence of the form $(1,3,1)$.

Conversely, as $A=\\{1,-1\\}$, so, it is easy to see that any sequence which
is $A$-equivalent to $(1,3,1)$ is an $A$-extremal sequence as this sequence
does not have any $A$-weighted zero-sum subsequence of consecutive terms. ∎

Remark: Theorem 2, Corollary 3 and Observation 1 together characterize the
$U(p)^{3}$-extremal sequences in $\mathbb{Z}_{p}$, when $p$ is a prime.

##  3 The case when $A=U(n)$

Let $m$ be a divisor of $n$. We will refer to the homomorphism
$\varphi:\mathbb{Z}_{n}\to\mathbb{Z}_{m}$ given by $a+n\mathbb{Z}\mapsto
a+m\mathbb{Z}$ as the natural map.

We use the notation $p^{r}\mid\mid n$ to mean that $p^{r}\mid n$ and that
$p^{r+1}\nmid n$. For a prime divisor $p$ of $n$, where $p^{r}\mid\mid n$, let
$A_{p}=U(p^{r})$ and let $S^{(p)}$ denote the sequence in $\mathbb{Z}_{p^{r}}$
which is the image of $S$ under the natural map
$\mathbb{Z}_{n}\to\mathbb{Z}_{p^{r}}$. The following was pointed out in
Observation 2.2 in [4]. We restate it here using our notation.

######  Observation 2.

Let $A=U(n)$ and let $S$ be a sequence in $\mathbb{Z}_{n}$. Then, $S$ is an
$A$-weighted zero-sum sequence in $\mathbb{Z}_{n}\iff$ for every prime divisor
$p$ of $n$, $S^{(p)}$ is an $A_{p}$-weighted zero-sum sequence in
$\mathbb{Z}_{p^{r}}$, where $p^{r}\mid\mid n$.

In fact, we have the following more general result, whose proof if similar to
the proof of the previous result. We use the notations in the second paragraph
of this section and we define $A_{p}^{j}=\\{\,x^{j}\mid x\in A_{p}\,\\}$.

######  Observation 3.

Let $A=U(n)^{j}$, where $j\geq 1$ and let $S$ be a sequence in
$\mathbb{Z}_{n}$. Then, $S$ is an $A$-weighted zero-sum sequence in
$\mathbb{Z}_{n}\iff$ for every prime divisor $p$ of $n$, $S^{(p)}$ is an
$A_{p}^{j}$-weighted zero-sum sequence in $\mathbb{Z}_{p^{r}}$, where
$p^{r}\mid\mid n$.

The following result is Lemma 4 in [5].

######  Lemma 1.

Let $S$ be a sequence in $\mathbb{Z}_{n}$ and let $p$ be a prime divisor of
$n$ which divides every element of $S$. Let $n^{\prime}=n/p$ and let
$S^{\prime}$ be the sequence in $\mathbb{Z}_{n^{\prime}}$ whose terms are
obtained by dividing the terms of $S$ by $p$. Let $A=U(n)$ and let
$A^{\prime}=U(n^{\prime})$. Suppose $S^{\prime}$ is an $A^{\prime}$-weighted
zero-sum sequence. Then $S$ is an $A$-weighted zero-sum sequence.

The following result is Lemma 2.1 (ii) in [4], which we restate here using our
terminology:

######  Lemma 2.

Let $A=U(p^{r})$, where $p$ is an odd prime and $r\geq 1$. If a sequence $S$
over $\mathbb{Z}_{p^{r}}$ has at least two terms coprime to $p$, then $S$ is
an $A$-weighted zero-sum sequence.

In Corollary 4 of [5], it was shown that for $A=U(n)$, where $n$ is odd, we
have $C_{A}(n)=2^{\Omega(n)}$. In the next theorem, we give a method to
construct $A$-extremal sequences in $\mathbb{Z}_{n}$.

######  Theorem 4.

Let $A=U(n)$, where $n$ is odd and let $p$ be a prime divisor of $n$. Let
$n^{\prime}=n/p$ and let $A^{\prime}=U(n^{\prime})$. Suppose
$S_{1}^{\prime}:(u_{1},\,\ldots,\,u_{k})$ and
$S_{2}^{\prime}:(v_{1},\,\ldots,\,v_{k})$ are $A^{\prime}$-extremal sequences
in $\mathbb{Z}_{n^{\prime}}$. Let $x^{*}\in\mathbb{Z}_{n}$ be an element which
is not divisible by $p$. Define a sequence $S$ in $\mathbb{Z}_{n}$ as
$S:(p\,u_{1},\,\ldots,\,p\,u_{k},\,x^{*},\,p\,v_{1},\,\ldots,\,p\,v_{k})$.
Then, $S$ is an $A$-extremal sequence in $\mathbb{Z}_{n}$.

###### Proof.

As $n^{\prime}$ is odd, so
$C_{A^{\prime}}(n^{\prime})=2^{\Omega(n^{\prime})}$. As $S_{1}^{\prime}$ is an
$A^{\prime}$-extremal sequence in $\mathbb{Z}_{n^{\prime}}$ of length $k$, so
$k=2^{\Omega(n^{\prime})}-1$. Let $S$ be defined as in the statement of the
theorem. We claim that $S$ does not have any $A$-weighted zero-sum subsequence
of consecutive terms. Suppose our claim is false. Let $T$ be an $A$-weighted
zero-sum subsequence of consecutive terms of $S$. Define sequences $S_{1}$ and
$S_{2}$ in $\mathbb{Z}_{n}$ as $S_{1}:(p\,u_{1},\,\ldots,\,p\,u_{k})$ and
$S_{2}:(p\,v_{1},\,\ldots,\,p\,v_{k})$.

Suppose $x^{*}$ is not a term of $T$. Then, $T$ is a subsequence of either
$S_{1}$ or $S_{2}$. Let $T^{\prime}$ be the sequence in
$\mathbb{Z}_{n^{\prime}}$ whose terms are obtained by dividing the terms of
$T$ by $p$. The image of $A$ under the natural map
$\mathbb{Z}_{n}\to\mathbb{Z}_{n^{\prime}}$ is contained in $A^{\prime}$. Then,
by dividing the $A$-weighted zero-sum which is obtained from $T$ by $p$, we
will get the contradiction that either $S_{1}^{\prime}$ or $S_{2}^{\prime}$
has an $A^{\prime}$-weighted zero-sum subsequence of consecutive terms, namely
$T^{\prime}$.

So, $x^{*}$ is a term of $T$. Now, considering the $A$-weighted zero-sum which
is obtained from $T$, we see that a multiple of $p$ is a unit multiple of
$x^{*}$. This gives the contradiction that $x^{*}$ is divisible by $p$. Thus,
our claim is true. As $n$ is odd, so $C_{A}(n)=2^{\Omega(n)}$. Also, as $S$ is
a sequence of length $2k+1=2^{\Omega(n)}-1$ in $\mathbb{Z}_{n}$, hence $S$ is
an $A$-extremal sequence in $\mathbb{Z}_{n}$. ∎

Let us see an example to see how we can use Theorem 4 to construct
$A$-extremal sequences. By Theorem 2, we have the $U(5)$-extremal sequences
$(2)$ and $(4)$ in $\mathbb{Z}_{5}$. Now, by Theorem 4, we get the
$U(25)$-extremal sequences $(10,4,20)$ and $(10,21,20)$ in $\mathbb{Z}_{25}$.
Repeating this process gives us the $U(75)$-extremal sequence
$(30,12,60,38,30,63,60)$ in $\mathbb{Z}_{75}$.

######  Lemma 3.

Let $A=U(n)$, where $n$ is odd and let $\ell=2^{\Omega(n)}-1$. Let $S$ be a
sequence in $\mathbb{Z}_{n}$ and let $q$ be a prime divisor of $n$. Suppose
$T$ is a subsequence of consecutive terms of $S$ of length at least
$(\ell+1)/2$ such that each term of $T$ is divisible by $q$. Then, $S$ has an
$A$-weighted zero-sum subsequence of consecutive terms.

###### Proof.

Let $n^{\prime}=n/q$ and let $A^{\prime}=U(n^{\prime})$. As
$\ell=2^{\Omega(n)}-1$, so $(\ell+1)/2=2^{\Omega(n)-1}$ and hence $T$ has
length at least $2^{\Omega(n^{\prime})}$. Let $T^{\prime}$ denote the sequence
in $\mathbb{Z}_{n^{\prime}}$ whose terms are obtained by dividing the terms of
$T$ by $q$. As $n$ is odd, so $n^{\prime}$ is odd. Now, as $T^{\prime}$ is a
sequence of length at least $2^{\Omega(n^{\prime})}$ in
$\mathbb{Z}_{n^{\prime}}$ and as
$C_{A^{\prime}}(n^{\prime})=2^{\Omega(n^{\prime})}$, so $T^{\prime}$ has an
$A^{\prime}$-weighted zero-sum subsequence of consecutive terms. Now, by Lemma
1, we see that $T$ has an $A$-weighted zero-sum subsequence of consecutive
terms. As $T$ is a subsequence of consecutive terms of $S$, hence $S$ has an
$A$-weighted zero-sum subsequence of consecutive terms. ∎

######  Corollary 4.

Let $A=U(n)$, where $n$ is odd. Suppose $S$ is an $A$-extremal sequence in
$\mathbb{Z}_{n}$ and $q$ is a prime divisor of $n$. Then, $q$ is coprime to at
least one term of $S$.

###### Proof.

As $S$ is an $A$-extremal sequence in $\mathbb{Z}_{n}$, so, if $S$ has length
$\ell$, then $\ell=2^{\Omega(n)}-1$. Suppose no term of $S$ is coprime to $q$.
Then, as $\ell$ is at least $(\ell+1)/2$, so by Lemma 3, we see that $S$ has
an $A$-weighted zero-sum subsequence of consecutive terms. This contradicts
our assumption about $S$. Hence, at least one term of $S$ is coprime to $q$. ∎

Theorem 5 shows that the method in Theorem 4 is the only way to obtain
$A$-extremal sequences in $\mathbb{Z}_{n}$, when $n$ is odd.

######  Theorem 5.

Let $A=U(n)$, where $n$ is odd. Suppose $S:(x_{1},\ldots,x_{\ell})$ is an
$A$-extremal sequence in $\mathbb{Z}_{n}$. Then, there is a prime divisor $p$
of $n$ such that $p$ divides all the terms of $S$ except the ‘middle’ term
$x_{k+1}$, where $k+1=(\ell+1)/2$.

Let $S_{1}:(x_{1},\ldots,x_{k})$ and $S_{2}:(x_{k+2},\ldots,x_{\ell})$. Let
$n^{\prime}=n/p$ and $A^{\prime}=U(n^{\prime})$. Let $S_{1}^{\prime}$ and
$S_{2}^{\prime}$ denote the sequences in $\mathbb{Z}_{n^{\prime}}$ which are
obtained by dividing the terms of $S_{1}$ and $S_{2}$ by $p$. Then,
$S_{1}^{\prime}$ and $S_{2}^{\prime}$ are $A^{\prime}$-extremal sequences in
$\mathbb{Z}_{n^{\prime}}$.

###### Proof.

As $S$ is an $A$-extremal sequence in $\mathbb{Z}_{n}$, so
$\ell=2^{\Omega(n)}-1$. Suppose for every prime divisor $q$ of $n$, there are
at least two terms of $S$ which are not divisible by $q$. Let $q$ be a prime
divisor of $n$ and let $q^{r}\mid\mid n$. Then, as $n$ is odd, so $q$ is odd.
Let $S^{(q)}$ be defined as at the beginning of this section. As the sequence
$S^{(q)}$ contains at least two units, so by Lemma 2, we see that $S^{(q)}$ is
an $A_{q}$-weighted zero-sum sequence, where $A_{q}=U(q^{r})$. As this is true
for every prime divisor $q$ of $n$, so by Observation 2, we see that $S$ is an
$A$-weighted zero-sum sequence. This contradicts our assumption about $S$. So,
there is a prime divisor $p$ of $n$, such that at most one term of $S$ is not
divisible by $p$. Now, by Corollary 4, we see that there is exactly one term
of $S$, say $x^{*}$, which is not divisible by $p$.

Suppose $x^{*}\neq x_{k+1}$, where $k+1=(\ell+1)/2$. Then, there is a
subsequence $T$ of consecutive terms of $S$ of length at least $k+1$ such that
$p$ divides every term of $T$. As $T$ has length at least $(\ell+1)/2$, so by
Lemma 3, we see that $S$ has an $A$-weighted zero-sum subsequence of
consecutive terms. This contradicts our assumption about $S$. Thus,
$x^{*}=x_{k+1}$.

Let $n^{\prime}=n/p$ and for $i=1,2$, let $S_{i}$ and $S_{i}^{\prime}$ be
defined as in the statement of the theorem. If $S_{1}^{\prime}$ has an
$A^{\prime}$-weighted zero-sum subsequence of consecutive terms, then by Lemma
1, we see that $S_{1}$ has an $A$-weighted zero-sum subsequence of consecutive
terms. As $S_{1}$ is a subsequence of consecutive terms of $S$, so $S$ has an
$A$-weighted zero-sum subsequence of consecutive terms. This contradicts our
assumption about $S$. Now, as $k+1=(\ell+1)/2=2^{\Omega(n)-1}$, so
$S_{1}^{\prime}$ has length $2^{\Omega(n^{\prime})}-1$ and hence
$S_{1}^{\prime}$ is an $A^{\prime}$-extremal sequence in
$\mathbb{Z}_{n^{\prime}}$. A similar argument shows that $S_{2}^{\prime}$ is
also an $A^{\prime}$-extremal sequence in $\mathbb{Z}_{n^{\prime}}$. ∎

Remark: Theorems 4 and 5 together characterize the $U(n)$-extremal sequences
in $\mathbb{Z}_{n}$, where $n$ is odd.

######  Observation 4.

Let $A=U(n)$, where $n$ is an odd integer and let $n=p_{1}p_{2}$ be a product
of $2$ not necessarily distinct primes. Suppose $S$ is an $A$-extremal
sequence in $\mathbb{Z}_{n}$. Then, $S$ is of the form
$(b_{1}q_{1},a_{1},b_{2}q_{1})$, where $q_{1}\nmid a_{1}$ and $q_{2}\nmid
b_{1}b_{2}$ and where $q_{1},q_{2}$ is a permutation of $p_{1},p_{2}$.

###### Proof.

As $S$ is an $A$-extremal sequence in $\mathbb{Z}_{n}$, so it has length $3$.
Let $S:(x_{1},x_{2},x_{3})$. By Theorem 5, there is a prime
$q_{1}\in\\{p_{1},p_{2}\\}$ such that $x_{2}$ is the only term of $S$ which is
coprime to $q_{1}$. Let $n^{\prime}:=n/q_{1}$, $x_{1}^{\prime}:=x_{1}/q_{1}$
and $x_{3}^{\prime}:=x_{3}/q_{1}$ and $A^{\prime}:=U(n^{\prime})$. Again, by
Theorem 5, we get that $(x_{1}^{\prime})$ and $(x_{3}^{\prime})$ are
$A^{\prime}$-extremal sequences in $\mathbb{Z}_{n^{\prime}}$. As $n^{\prime}$
is a prime, say $q_{2}$, so $x_{1}^{\prime},x_{3}^{\prime}\in U(q_{2})$. Thus,
$S$ is of the form $(b_{1}q_{1},a_{1},b_{2}q_{1})$, where $q_{1}\nmid a_{1}$
and $q_{2}\nmid b_{1}b_{2}$ and where $q_{1},q_{2}$ is a permutation of
$p_{1},p_{2}$. ∎

######  Observation 5.

Let $A=U(n)$, where $n$ is an odd integer and let $n=p_{1}p_{2}p_{3}$ be a
product of $3$ not necessarily distinct primes. Suppose $S$ is an $A$-extremal
sequence in $\mathbb{Z}_{n}$. Then, it has length $7$. By a similar argument,
using Theorem 5 as in the previous paragraph, we get that $S$ can have two
possible forms. In the following two sequences, $q_{1},q_{2},q_{3}$ is a
permutation of $p_{1},p_{2},p_{3}$ and for all values of $i$, we have
$a_{i},b_{i},c_{i}\in\mathbb{Z}_{n}$.

$(a_{1}q_{1}q_{2},b_{1}q_{1},a_{2}q_{1}q_{2},c_{1},a_{3}q_{1}q_{2},b_{2}q_{1},a_{4}q_{1}q_{2})$,
$q_{1}\nmid c_{1}$, $q_{2}\nmid b_{1}b_{2}$ and $q_{3}\nmid
a_{1}a_{2}a_{3}a_{4}$.

$(a_{1}q_{1}q_{2},b_{1}q_{1},a_{2}q_{1}q_{2},c_{1},b_{2}q_{1}q_{3},a_{3}q_{1},b_{3}q_{1}q_{3})$,
$q_{1}\nmid c_{1}$, $q_{2}\nmid b_{1}b_{2}b_{3}$ and $q_{3}\nmid
a_{1}a_{2}a_{3}$.

##  4 The case when $A=U(n)^{2}$

We need the following result which is Lemma 1 in [3].

######  Lemma 4.

Let $A=U(n)^{2}$, where $n=p^{r}$ and $p\geq 7$ is a prime. Let
$x_{1},x_{2},x_{3}\in U(n)$. Then, $Ax_{1}+Ax_{2}+Ax_{3}=\mathbb{Z}_{n}$.

The next result is Corollary 6 of [5], which is an easy consequence of Lemma
4.

######  Corollary 5.

Let $A=U(n)^{2}$, where $n=p^{r}$ and $p\geq 7$ is a prime. Let $S$ be a
sequence in $\mathbb{Z}_{n}$ such that at least three terms of $S$ are in
$U(n)$. Then, $S$ is an $A$-weighted zero-sum sequence.

Remark: The conclusion of Corollary 5 may not hold when $p<7$. For $n=2,5$,
the sequence $(1,1,1)$ in $\mathbb{Z}_{n}$ is not a $U(n)^{2}$-weighted zero-
sum sequence. The sequence $(1,2,1)$ in $\mathbb{Z}_{3}$ is not a
$U(3)^{2}$-weighted zero-sum sequence.

The next result is Lemma 8 in [5].

######  Lemma 5.

Let $S$ be a sequence in $\mathbb{Z}_{n}$ and let $p$ be a prime divisor of
$n$ which divides every element of $S$. Let $n^{\prime}=n/p$ and let
$S^{\prime}$ be the sequence in $\mathbb{Z}_{n^{\prime}}$ whose terms are
obtained by dividing the terms of $S$ by $p$. Let $A=U(n)^{2}$ and let
$A^{\prime}=U(n^{\prime})^{2}$. Suppose $S^{\prime}$ is an
$A^{\prime}$-weighted zero-sum sequence. Then, $S$ is an $A$-weighted zero-sum
sequence.

In Corollary 7 of [5], it was shown that for $A=U(n)^{2}$, where every prime
divisor of $n$ is at least 7, we have $C_{A}(n)=3^{\Omega(n)}$. We now give a
method to construct $A$-extremal sequences in $\mathbb{Z}_{n}$.

######  Theorem 6.

Let $A=U(n)^{2}$ and let $p$ be a prime divisor of $n$. Let $n^{\prime}=n/p$
and let $A^{\prime}=U(n^{\prime})^{2}$. Let
$S_{1}^{\prime}:(u_{1},\,\ldots,\,u_{k}),~{}S_{2}^{\prime}:(v_{1},\,\ldots,\,v_{k})$
and $S_{3}^{\prime}:(w_{1},\,\ldots,\,w_{k})$ be $A^{\prime}$-extremal
sequences in $\mathbb{Z}_{n^{\prime}}$.

Let $x^{*},x^{**}\in\mathbb{Z}_{n}$ such that the image of the sequence
$(x^{*},x^{**})$ under the natural map $\mathbb{Z}_{n}\to\mathbb{Z}_{p}$ does
not have any $Q_{p}$-weighted zero-sum subsequence. Let
$S:(p\,u_{1},\,\ldots,\,p\,u_{k},\,x^{*},\,p\,v_{1},\,\ldots,\,p\,v_{k},\,x^{**},\,p\,w_{1},\,\ldots,\,p\,w_{k})$.
Then, $S$ is an $A$-extremal sequence in $\mathbb{Z}_{n}$.

###### Proof.

Since $S_{1}^{\prime}$ is an $A^{\prime}$-extremal sequence of length $k$ in
$\mathbb{Z}_{n^{\prime}}$, so $k=3^{\Omega(n^{\prime})}-1$. Let $S$ be defined
as in the statement of the theorem. We claim that $S$ does not have any
$A$-weighted zero-sum subsequence of consecutive terms. Suppose our claim is
false. Let $T$ be an $A$-weighted zero-sum subsequence of consecutive terms of
$S$. Define sequences $S_{1},S_{2}$ and $S_{3}$ in $\mathbb{Z}_{n}$ as
$S_{1}:(p\,u_{1},\,\ldots,\,p\,u_{k}),~{}S_{2}:(p\,v_{1},\,\ldots,\,p\,v_{k})$,
$S_{3}:(p\,w_{1},\,\ldots,\,p\,w_{k})$.

The image of $A$ under the natural map
$\mathbb{Z}_{n}\to\mathbb{Z}_{n^{\prime}}$ is contained in $A^{\prime}$. So,
if $T$ is a subsequence of either $S_{1}$, $S_{2}$ or $S_{3}$, then, as we
have seen in the second paragraph of the proof of Theorem 4, we will get the
contradiction that either $S_{1}^{\prime}$, $S_{2}^{\prime}$ or
$S_{3}^{\prime}$ has an $A^{\prime}$-weighted zero-sum subsequence of
consecutive terms. So, $T\cap\\{x^{*},x^{**}\\}\neq\varnothing$. Now,
considering the $A$-weighted zero-sum which is obtained from $T$, we see that
an $A$-weighted linear combination of a subsequence of $(x^{*},x^{**})$ is a
multiple of $p$.

The image of $A$ under the natural map $\mathbb{Z}_{n}\to\mathbb{Z}_{p}$ is
contained in $Q_{p}$. So, we get a contradiction that the image of the
sequence $(x^{*},x^{**})$ under the natural map
$\mathbb{Z}_{n}\to\mathbb{Z}_{p}$ has a $Q_{p}$-weighted zero-sum subsequence.
Thus, our claim is true. Now, as $S$ is a sequence of length
$3k+2=3^{\Omega(n)}-1$ in $\mathbb{Z}_{n}$, hence $S$ is an $A$-extremal
sequence in $\mathbb{Z}_{n}$. ∎

We can use Theorem 6 to construct $A$-extremal sequences in $\mathbb{Z}_{n}$.
For example, from Corollary 2, we get the $U(5)^{2}$-extremal sequences
$(4,2)$, $(1,3)$ and $(3,4)$ in $\mathbb{Z}_{5}$. Now, by using these, we get
the $U(25)^{2}$-extremal sequence $(20,10,21,5,15,12,15,20)$ in
$\mathbb{Z}_{25}$.

Theorems 7 and 8 will show us that the method in Theorem 6 is the only way to
obtain $A$-extremal sequences in $\mathbb{Z}_{n}$, when any prime divisor of
$n$ is at least $7$. We shall need the following lemmas.

######  Lemma 6.

Let $A=U(n)^{2}$, where every prime divisor of $n$ is at least 7 and let
$\ell=3^{\Omega(n)}-1$. Let $S$ be a sequence in $\mathbb{Z}_{n}$ and let $q$
be a prime divisor of $n$. Suppose we get a subsequence, say $T$, of
consecutive terms of $S$ of length at least $(\ell+1)/3$ such that each term
of $T$ is divisible by $q$. Then, $S$ has an $A$-weighted zero-sum subsequence
of consecutive terms.

###### Proof.

Let $n^{\prime}=n/q$ and let $T^{\prime}$ denote the sequence in
$\mathbb{Z}_{n^{\prime}}$ whose terms are obtained by dividing the terms of
$T$ by $q$. As $T$ is a sequence of length at least $(\ell+1)/3$ in
$\mathbb{Z}_{n}$, so $T^{\prime}$ is a sequence of length at least
$3^{\Omega(n^{\prime})}$ in $\mathbb{Z}_{n^{\prime}}$. Also, as every prime
divisor of $n^{\prime}$ is at least 7 and as
$C_{A^{\prime}}(n^{\prime})=3^{\Omega(n^{\prime})}$, so, $T^{\prime}$ has an
$A^{\prime}$-weighted zero-sum subsequence of consecutive terms, where
$A^{\prime}=U(n^{\prime})^{2}$. Now, by Lemma 5, we see that $T$ has an
$A$-weighted zero-sum subsequence of consecutive terms. As $T$ is a
subsequence of consecutive terms of $S$, hence $S$ has an $A$-weighted zero-
sum subsequence of consecutive terms. ∎

######  Corollary 6.

Let $A=U(n)^{2}$, where every prime divisor of $n$ is at least 7. Suppose $S$
is an $A$-extremal sequence in $\mathbb{Z}_{n}$. Then, for any prime divisor
$q$ of $n$, we have that $q$ is coprime to at least two terms of $S$.

###### Proof.

As $S$ is an $A$-extremal sequence in $\mathbb{Z}_{n}$, so, if $S$ has length
$\ell$, then $\ell=3^{\Omega(n)}-1$. Suppose there is a prime divisor $q$ of
$n$ such that at most one term of $S$ is coprime to $q$. Then we get a
subsequence, say, $T$ of consecutive terms of $S$ of length at least
$(\ell+1)/3$ such that each term of $T$ is divisible by $q$. Now, by Lemma 6,
we see that $S$ has an $A$-weighted zero-sum subsequence of consecutive terms.
This contradicts our assumption about $S$. Hence, at least two terms of $S$
are coprime to $q$. ∎

######  Theorem 7.

Let $A=U(n)^{2}$, where any prime divisor of $n$ is at least $7$. Suppose
$S:(x_{1},\ldots,x_{\ell})$ is an $A$-extremal sequence in $\mathbb{Z}_{n}$.
Then, there is a prime divisor $p$ of $n$ such that $p$ divides all the terms
of $S$ except the terms $x_{k+1}$ and $x_{2(k+1)}$, where $k+1=(\ell+1)/3$.

Let $S_{1}:(x_{1},\ldots,x_{k})$, $S_{2}:(x_{k+2},\ldots,x_{2k+1})$ and
$S_{3}:(x_{2k+3},\ldots,x_{\ell})$. Let $n^{\prime}=n/p$ and
$A^{\prime}=U(n^{\prime})^{2}$. Let $S_{1}^{\prime}$, $S_{2}^{\prime}$ and
$S_{3}^{\prime}$ denote the sequences in $\mathbb{Z}_{n}^{\prime}$ which are
obtained by dividing the terms of $S_{1}$, $S_{2}$ and $S_{3}$ by $p$. Then,
$S_{1}^{\prime}$, $S_{2}^{\prime}$ and $S_{3}^{\prime}$ are
$A^{\prime}$-extremal sequences in $\mathbb{Z}_{n^{\prime}}$.

###### Proof.

As $S$ is an $A$-extremal sequence in $\mathbb{Z}_{n}$, so
$\ell=3^{\Omega(n)}-1$. Suppose for any prime divisor $q$ of $n$, there are at
least three terms of $S$ which are not divisible by $q$. Let $q$ be a prime
divisor of $n$, $q^{r}\mid\mid n$ and let $A_{q}=U(q^{r})$. Then, $S^{(q)}$
has at least three units, where $S^{(q)}$ is as defined before Observation 2.
By Corollary 5, as $q$ is at least 7, so $S^{(q)}$ is an $A_{q}^{2}$-weighted
zero-sum sequence in $\mathbb{Z}_{q^{r}}$. As this is true for every prime
divisor $q$ of $n$, so by Observation 3, we see that $S$ is an $A$-weighted
zero-sum sequence. This contradicts our assumption about $S$. So we see that,
there is a prime divisor $p$ of $n$, such that at most two terms of $S$ are
not divisible by $p$. Now, by using Corollary 6, we see that exactly two terms
of $S$, say $x^{*}$ and $x^{**}$, are not divisible by $p$.

Let us assume that $x^{*}$ occurs before $x^{**}$ in $S$. Suppose $x^{*}\neq
x_{k+1}$ or $x^{**}\neq x_{2(k+1)}$, where $k+1=(\ell+1)/3$. Then, there is a
subsequence $T$ of consecutive terms of $S$ of length at least $k+1$ such that
$p$ divides every term of $T$. As every prime divisor of $n$ is at least 7 and
as $T$ has length at least $(\ell+1)/3$, so by Lemma 6, we see that $S$ has an
$A$-weighted zero-sum subsequence of consecutive terms. This contradicts our
assumption about $S$. Thus, $x^{*}=x_{k+1}$ and $x^{**}=x_{2(k+1)}$.

Let $n^{\prime}=n/p$ and for $1\leq i\leq 3$, let $S_{i}$ and $S_{i}^{\prime}$
be defined as in the statement of the theorem. If $S_{1}^{\prime}$ has an
$A^{\prime}$-weighted zero-sum subsequence of consecutive terms, then by Lemma
5, we see that $S_{1}$ has an $A$-weighted zero-sum subsequence of consecutive
terms. As $S_{1}$ is a subsequence of consecutive terms of $S$, so $S$ has an
$A$-weighted zero-sum subsequence of consecutive terms. This contradicts our
assumption about $S$.

As $k+1=(\ell+1)/3$ and as $\ell+1=3^{\Omega(n)}$, so
$k+1=3^{\Omega(n^{\prime})}$. Thus, as $S_{1}^{\prime}$ has length $k$, so
$S_{1}^{\prime}$ has to be an $A^{\prime}$-extremal sequence in
$\mathbb{Z}_{n^{\prime}}$. Similar arguments show that $S_{2}^{\prime}$ and
$S_{3}^{\prime}$ are also $A^{\prime}$-extremal sequences in
$\mathbb{Z}_{n^{\prime}}$. ∎

######  Lemma 7.

Let $m$ be a divisor of $n$ and let $b\in U(m)$. Then $\exists~{}a\in U(n)$
such that $a\mapsto b$ under the natural map
$\mathbb{Z}_{n}\to\mathbb{Z}_{m}$. If $b\in U(m)^{2}$, then we can choose
$a\in U(n)^{2}$.

###### Proof.

Let $\mathbb{Z}^{+}=\\{i\in\mathbb{Z}\mid i>0\\}$. As $b$ is coprime to $m$,
so by Dirichlet’s theorem (see [6], for instance), there are infinitely many
primes in the arithmetic progression $\\{\,b+im\mid i\in\mathbb{Z}^{+}\,\\}$.
As only finitely many of these primes will divide $n$, we see that
$\exists~{}i\in\mathbb{Z}^{+}$ such that $a=b+im\in U(n)$. Also, $a\mapsto b$
under the natural map $\mathbb{Z}_{n}\to\mathbb{Z}_{m}$. The proof when $b\in
U(m)^{2}$ now follows from this. ∎

######  Theorem 8.

We continue with the notations which were used in the statement and proof of
Theorem 7. Then, the image of the sequence $(x^{*},x^{**})$ under the natural
map $\mathbb{Z}_{n}\to\mathbb{Z}_{p}$ is a $Q_{p}$-extremal sequence in
$\mathbb{Z}_{p}$.

###### Proof.

If $n=p$, then $S$ is a $Q_{p}$-extremal sequence in $\mathbb{Z}_{p}$. So we
can assume that $n$ is not a prime. Suppose the image of the sequence
$(x^{*},x^{**})$ under the natural map $\mathbb{Z}_{n}\to\mathbb{Z}_{p}$ is a
$Q_{p}$-weighted zero-sum sequence. Then, by Lemma 7, $\exists~{}a,b\in
U(n)^{2}$ such that $ax^{*}+bx^{**}\in\mathbb{Z}_{n}$ maps to
$0\in\mathbb{Z}_{p}$ under the natural map $\mathbb{Z}_{n}\to\mathbb{Z}_{p}$.
Thus, $p$ divides $y=ax^{*}+bx^{**}$ in $\mathbb{Z}_{n}$. Let $T$ be the
sequence which is obtained by removing the terms $x^{*}$ and $x^{**}$ from $S$
and by adding the term $y$ at the end of $S$. Now, by using Theorem 7, we see
that $p$ divides all the terms of $T$.

Let $n^{\prime}=n/p$ and let $A^{\prime}=U(n^{\prime})^{2}$. Consider the
sequence $T^{\prime}$ in $\mathbb{Z}_{n^{\prime}}$ whose terms are obtained by
dividing the terms of $T$ by $p$. Let $q$ be a prime divisor of $n^{\prime}$.
As $S_{1}^{\prime}$ is an $A^{\prime}$-extremal sequence in
$\mathbb{Z}_{n^{\prime}}$ and as every prime divisor of $n^{\prime}$ is at
least 7, so by Corollary 6, at least two terms of $S_{1}^{\prime}$ are coprime
to $q$. Similarly, at least two terms of $S_{2}^{\prime}$ are coprime to $q$.
Hence, $T^{\prime}$ has at least 4 terms which are coprime to $q$. As this is
true for any prime divisor $q$ of $n^{\prime}$, by the same argument as in the
first paragraph of the proof of Theorem 7, we see that $T^{\prime}$ is an
$A^{\prime}$-weighted zero-sum sequence in $\mathbb{Z}_{n^{\prime}}$. Now, by
Lemma 5, we see that $T$ is an $A$-weighted zero-sum sequence in
$\mathbb{Z}_{n}$.

Now, as $T$ is also a concatenation of the sequences $S_{1},~{}S_{2},~{}S_{3}$
and $(ax^{*}+bx^{**})$ and as $a,b\in A$, so $S$ is an $A$-weighted zero-sum
sequence in $\mathbb{Z}_{n}$. This contradicts our assumption about $S$. Thus,
the image of the sequence $(x^{*},x^{**})$ under the natural map
$\mathbb{Z}_{n}\to\mathbb{Z}_{p}$ cannot be a $Q_{p}$-weighted zero-sum
sequence in $\mathbb{Z}_{p}$. So, it is a $Q_{p}$-extremal sequence in
$\mathbb{Z}_{p}$, as $C_{Q_{p}}(p)=3$ for $p>2$. ∎

Remark: Theorems 6, 7 and 8 together characterize the $U(n)^{2}$-extremal
sequences in $\mathbb{Z}_{n}$, where any prime divisor of $n$ is at least $7$.

We conclude this section with the following results about these sequences.

######  Lemma 8.

We continue with the notations which were used in the statement of Theorem 7.
Then, $p$ is the unique prime divisor of $n$ such that $p$ is coprime to
exactly two terms of $S$ and any other prime divisor $q$ of $n$ is coprime to
at least three terms of $S$.

###### Proof.

By Theorem 7, we know that $p$ is coprime to exactly two terms of $S$. Let $q$
be a prime divisor of $n$ other than $p$. By Corollary 6, $q$ has to be
coprime to at least two terms of $S$. We claim that $q$ cannot be coprime to
exactly two terms of $S$. Suppose our claim is false. Then, $q$ is coprime to
exactly two terms of $S$. Now, by a similar argument as in the second
paragraph of the proof of Theorem 7, we see that those two terms have to be
$x^{*}$ and $x^{**}$. So, $q$ divides all the terms of $S_{1}$. As $q\neq p$,
this implies that $q$ divides all the terms of $S_{1}^{\prime}$. However, as
$q$ is a divisor of $n^{\prime}$ and as $S_{1}^{\prime}$ is an
$A^{\prime}$-extremal sequence, so by Corollary 6, $q$ has to be coprime to at
least two terms of $S_{1}^{\prime}$. Thus, we get a contradiction. Hence, our
claim must be true. ∎

######  Observation 6.

Let $A=U(n)^{2}$, where $n=p_{1}p_{2}$ is a product of two not necessarily
distinct primes which are at least 7. Then, $S$ is an $A$-extremal sequence in
$\mathbb{Z}_{n}$ if and only if $S$ is of the form

$(b_{1}q_{1},b_{2}q_{1},a_{1},b_{3}q_{1},b_{4}q_{1},a_{2},b_{5}q_{1},b_{6}q_{1}),$

where $q_{1},q_{2}$ is a permutation of $p_{1},p_{2}$, $(b_{i},b_{i+1})$ is a
$Q_{q_{{}_{2}}}$-extremal sequence in $\mathbb{Z}_{q_{{}_{2}}}$ for $i=1,3,5$
and the image of $(a_{1},a_{2})$ under the natural map
$\mathbb{Z}_{n}\to\mathbb{Z}_{q_{{}_{1}}}$ is a $Q_{q_{{}_{1}}}$-extremal
sequence in $\mathbb{Z}_{q_{{}_{1}}}$.

###### Proof.

As $S$ is an $A$-extremal sequence in $\mathbb{Z}_{n}$, so it has length $8$.
Let $S:(x_{1},\ldots,x_{8})$. By Theorem 7, there is a prime
$q_{1}\in\\{p_{1},p_{2}\\}$ such that $x_{3},x_{6}$ are the only terms of $S$
which are coprime to $q_{1}$. Let $n^{\prime}:=n/q_{1}$,
$A^{\prime}:=U(n^{\prime})^{2}$ and let
$S_{1}^{\prime}:(x_{1}/q_{1},x_{2}/q_{1})$,
$S_{2}^{\prime}:(x_{4}/q_{1},x_{5}/q_{1})$,
$S_{3}^{\prime}:(x_{7}/q_{1},x_{8}/q_{1})$.

By Theorem 7, we also get that $S_{1}^{\prime}$, $S_{2}^{\prime}$ and
$S_{3}^{\prime}$ are $A^{\prime}$-extremal sequences in
$\mathbb{Z}_{n^{\prime}}$ and so they are $Q_{q_{{}_{2}}}$-extremal sequences
in $\mathbb{Z}_{q_{{}_{2}}}$, where $q_{2}$ is the prime $n^{\prime}$. By
Theorem 8, we get that the image of $(x_{3},x_{6})$ under the natural map
$\mathbb{Z}_{n}\to\mathbb{Z}_{q_{{}_{1}}}$ is a $Q_{q_{{}_{1}}}$-extremal
sequence in $\mathbb{Z}_{q_{{}_{1}}}$. Hence, we see that $S$ is of the form
as mentioned in the statement of the observation. Theorem 6, shows that any
such sequence will be an $A$-extremal sequence in $\mathbb{Z}_{n}$. ∎

##  5 The case when $A=U(n)^{3}$

Let $n=p_{1}p_{2}\ldots p_{s}$ be squarefree. Let $n_{1}$ be the divisor of
$n$ such that for any prime divisor $p$ of $n$, we have $p\mid n_{1}\iff
p\equiv 1~{}(mod~{}3)$. Also, let $n_{2}=n/n_{1}$. We will follow the notation
$n=n_{1}n_{2}$ for the rest of this section.

The following result is Lemma 9 in [5].

######  Lemma 9.

Let $A=U(p)^{3}$, where $p$ is a prime and $p\neq 2,7,13$. Let $S$ be a
sequence in $\mathbb{Z}_{p}$ such that at least three elements of $S$ are in
$U(p)$. Then $S$ is an $A$-weighted zero-sum sequence.

Remark: We can check that the sequence $(1,1,1)$ in $\mathbb{Z}_{p}$ is not a
$U(p)^{3}$-weighted zero-sum sequence when $p=2,7,13$. So, the conclusion of
Lemma 9 is not true for these values of $p$ (as has been mentioned in [5], in
the Remark after Lemma 9).

In Corollary 9 of [5], it was shown that for $A=U(n)^{3}$, where $n$ is
squarefree and is not divisible by 2, 7 and 13, we have
$C_{A}(n)=2^{\Omega(n_{2})}3^{\Omega(n_{1})}$. We now give a method to
construct $A$-extremal sequences in $\mathbb{Z}_{n}$.

######  Theorem 9.

Let $A=U(n)^{3}$, where $n$ is squarefree and is not divisible by 2,7 and 13.
Let $p$ be a prime divisor of $n$. Let $n^{\prime}=n/p$ and let
$A^{\prime}=U(n^{\prime})^{3}$. Then, the following constructions give us an
$A$-extremal sequence $S$ in $\mathbb{Z}_{n}$ in each of the following cases:

Case: $p\equiv 1~{}(mod~{}3)$

Let
$S_{1}^{\prime}:(u_{1},\,\ldots,\,u_{k}),~{}S_{2}^{\prime}:(v_{1},\,\ldots,\,v_{k})$
and $S_{3}^{\prime}:(w_{1},\,\ldots,\,w_{k})$ be $A^{\prime}$-extremal
sequences in $\mathbb{Z}_{n^{\prime}}$. Let $x^{*},x^{**}\in\mathbb{Z}_{n}$
such that the image of the sequence $(x^{*},x^{**})$ under the natural map
$\mathbb{Z}_{n}\to\mathbb{Z}_{p}$ does not have any $U(p)^{3}$-weighted zero-
sum subsequence. Consider the sequence $S$ in $\mathbb{Z}_{n}$ defined as
$S:(p\,u_{1},\,\ldots,\,p\,u_{k},\,x^{*},\,p\,v_{1},\,\ldots,\,p\,v_{k},\,x^{**},\,p\,w_{1},\,\ldots,\,p\,w_{k})$.

Case: $p\not\equiv 1~{}(mod~{}3)$

Let
$S_{1}^{\prime}:(u_{1},\,\ldots,\,u_{k}),~{}S_{2}^{\prime}:(v_{1},\,\ldots,\,v_{k})$
be $A^{\prime}$-extremal sequences in $\mathbb{Z}_{n^{\prime}}$. Let
$x^{*}\in\mathbb{Z}_{n}$ such that it is not divisible by $p$. Consider the
sequence $S$ in $\mathbb{Z}_{n}$ defined as
$S:(p\,u_{1},\,\ldots,\,p\,u_{k},\,x^{*},\,p\,v_{1},\,\ldots,\,p\,v_{k})$.

###### Proof.

The proof of Theorem 9 is very similar to the proofs of Theorem 4 and Theorem
6. ∎

Let us see two examples on how we can use Theorem 9 to construct $A$-extremal
sequences.

As $U(5)^{3}=U(5)$, so, by Theorem 2, we have the $U(5)^{3}$-extremal
sequences $(2),(2)$ and $(4)$ in $\mathbb{Z}_{5}$. Also, by Corollary 3,
$(-1,2)$ is a $U(19)^{3}$-extremal sequence in $\mathbb{Z}_{19}$. So, by
Theorem 9, we get the $U(95)^{3}$-extremal sequence $(38,37,38,78,76)$ in
$\mathbb{Z}_{95}$.

By Corollary 3, $(3,6)$, $(5,7)$ are $U(19)^{3}$-extremal sequences in
$\mathbb{Z}_{19}$. So, by Theorem 9, we get the $U(95)^{3}$-extremal sequence
$(15,30,69,25,35)$ in $\mathbb{Z}_{95}$.

We can prove a similar result like Lemma 3 and Lemma 6, for $A=U(n)^{3}$,
whose proof is also similar to the proofs of those lemmas.

######  Lemma 10.

Let $A=U(n)^{3}$, where $n$ is squarefree and is not divisible by 2, 7 and 13
and let $\ell=2^{\Omega(n_{2})}3^{\Omega(n_{1})}-1$. Let $S$ be a sequence in
$\mathbb{Z}_{n}$ and let $q$ be a prime divisor of $n$.

Then, $S$ has an $A$-weighted zero-sum subsequence of consecutive terms if $q$
divides $n_{1}$ (resp. $n_{2}$) and we get a subsequence, say $T$, of
consecutive terms of $S$ of length at least $(\ell+1)/3$ (resp. $(\ell+1)/2$)
such that each term of $T$ is divisible by $q$.

The proof of the next corollary is very similar to the proofs of Corollaries 4
and 6.

######  Corollary 7.

Let $A=U(n)^{3}$, where $n$ is squarefree and is not divisible by 2, 7 and 13.
Suppose $S$ is an $A$-extremal sequence in $\mathbb{Z}_{n}$. Then, for any
prime divisor $q$ of $n_{1}$ (resp. of $n_{2}$), we have that $q$ is coprime
to at least two terms (resp. at least one term) of $S$.

######  Theorem 10.

Let $A=U(n)^{3}$, where $n$ is squarefree and is not divisible by 2, 7 and 13.
Suppose $S:(x_{1},\ldots,x_{\ell})$ is an $A$-extremal sequence in
$\mathbb{Z}_{n}$. Then, there are two possible cases:

Case: There is a prime divisor $p$ of $n_{1}$ such that $p$ divides all the
terms of $S$ except the terms $x_{k+1}$ and $x_{2(k+1)}$, where
$k+1=(\ell+1)/3$.

Let $S_{1}:(x_{1},\ldots,x_{k})$, $S_{2}:(x_{k+2},\ldots,x_{2k+1})$ and
$S_{3}:(x_{2k+3},\ldots,x_{\ell})$. Let $n^{\prime}=n/p$ and
$A^{\prime}=U(n^{\prime})^{3}$. Let $S_{1}^{\prime}$, $S_{2}^{\prime}$ and
$S_{3}^{\prime}$ denote the sequences in $\mathbb{Z}_{n}^{\prime}$ which are
obtained by dividing the terms of $S_{1}$, $S_{2}$ and $S_{3}$ by $p$.

Then, $S_{1}^{\prime}$, $S_{2}^{\prime}$ and $S_{3}^{\prime}$ are
$A^{\prime}$-extremal sequences in $\mathbb{Z}_{n^{\prime}}$. Also, the image
of the sequence $(x_{k+1},x_{2k+2})$ under the natural map
$\mathbb{Z}_{n}\to\mathbb{Z}_{p}$ is a $U(p)^{3}$-extremal sequence in
$\mathbb{Z}_{p}$.

Case: There is a prime divisor $p$ of $n_{2}$ such that $p$ divides all the
terms of $S$ except the term $x_{k+1}$, where $k+1=(\ell+1)/2$.

Let $S_{1}:(x_{1},\ldots,x_{k})$ and $S_{2}:(x_{k+2},\ldots,x_{\ell})$. Let
$n^{\prime}=n/p$ and $A^{\prime}=U(n^{\prime})$. Let $S_{1}^{\prime}$ and
$S_{2}^{\prime}$ denote the sequences in $\mathbb{Z}_{n^{\prime}}$ which are
obtained by dividing the terms of $S_{1}$ and $S_{2}$ by $p$.

Then, $S_{1}^{\prime}$ and $S_{2}^{\prime}$ are $A^{\prime}$-extremal
sequences in $\mathbb{Z}_{n^{\prime}}$.

###### Proof.

As $S$ is an $A$-extremal sequence in $\mathbb{Z}_{n}$, so
$\ell=2^{\Omega(n_{2})}3^{\Omega(n_{1})}-1$.

Case: For any prime divisor $q$ of $n_{1}$, at least three terms of $S$ are
coprime to $q$ and for any prime divisor $q$ of $n_{2}$, at least two terms of
$S$ are coprime to $q$.

As $q\neq 2,7,13$, so if $q$ divides $n_{1}$, then by Lemma 9, $S^{(q)}$ is an
$U(q)^{3}$-weighted zero-sum sequence in $\mathbb{Z}_{q}$. As $q$ is an odd
prime, so if $q$ divides $n_{2}$, then by Lemma 2, we see that $S^{(q)}$ is an
$U(q)$-weighted zero-sum sequence in $\mathbb{Z}_{q}$.

So, for any prime divisor $q$ of $n$, we get that $S^{(q)}$ is an
$U(q)^{3}$-weighted zero-sum sequence in $\mathbb{Z}_{q}$ and hence by
Observation 3, we see that $S$ is an $A$-weighted zero-sum sequence. This
contradicts our assumption about $S$.

Case: There is a prime divisor $p$ of $n_{1}$ such that at most two terms of
$S$ are coprime to $p$.

The proof of the result in this case is very similar to the proofs of Theorems
7 and 8, where we use Lemma 9, Lemma 10 and Corollary 7 in place of Corollary
5, Lemma 6 and Corollary 6. We also use a consequence of Lemma 7.

Case: There is a prime divisor $p$ of $n_{2}$ such that at most one term of
$S$ is coprime to $p$.

The proof of the result in this case is very similar to the proof of Theorem
5, where we use Lemma 10 and Corollary 7 in place of Lemma 3 and Corollary 4.
∎

Remark: Theorems 9 and 10 together characterize the $U(n)^{3}$-extremal
sequences in $\mathbb{Z}_{n}$, where $n$ is squarefree and is not divisible by
2, 7 and 13.

##  6 Concluding remarks:

In Theorem 5, we have classified the $U(n)$-weighted extremal sequences in
$\mathbb{Z}_{n}$, when $n$ is odd. The corresponding result for an even
integer could be studied. In the characterization of $U(n)^{3}$-weighted
extremal sequences in $\mathbb{Z}_{n}$, it will be interesting if we could
remove the assumption that $n$ is squarefree.

In Lemma 7, we have proved that the image of $U(n)$ under the natural map
$\mathbb{Z}_{n}\to\mathbb{Z}_{m}$ is $U(m)$ using Dirichlet’s theorem. In [3],
an elementary proof of this fact is given just before Section 3.

In Definition 6, if a sequence $T$ is obtained by a permutation of the terms
of a sequence $S$, then we do not say that $T$ is $A$-equivalent to $S$. This
is because, if $S$ has an $A$-weighted zero-sum subsequence of consecutive
terms, then the same may not be true for $T$. This is why our definition
differs from the one given at the end of Section 2 in [2].

Acknowledgement. Santanu Mondal would like to acknowledge CSIR, Govt. of
India, for a research fellowship.

## References

  * [1] S. D. Adhikari, Y. G. Chen, Davenport constant with weights and some related questions, II, J. Combin. Theory A, 115, No. 1, (2008), 178-184, https://doi.org/10.1016%2Fj.jcta.2007.03.004
  * [2] S. D. Adhikari, I. Molla, S. Paul, Extremal Sequences for Some Weighted Zero-Sum Constants for Cyclic Groups, Combinatorial and Additive Number Theory IV, Springer Proc. in Math. & Stat., 347, (2021), 1-10,
https://doi.org/10.1007/978-3-030-67996-5_1

  * [3] M. N. Chintamani, B. K. Moriya, Generalizations of some zero sum theorems, Proc. Indian Acad. Sci. (Math. Sci.), 122, No. 1, (2012), 15-21, https://doi.org/10.1007%2Fs12044-012-0058-7
  * [4] Simon Griffiths, The Erdős-Ginzberg-Ziv theorem with units, Discrete Mathematics, 308, No. 23, (2008), 5473-5484,
https://doi.org/10.1016%2Fj.disc.2007.09.060

  * [5] S. Mondal, K. Paul, S. Paul, On a different weighted zero-sum constant, submitted and e-print is available at http://arxiv.org/abs/2110.02539
  * [6] Atle Selberg, An Elementary Proof of Dirichlet’s Theorem about Primes in an Arithmetic Progression, Annals of Mathematics, 50, No. 2, (1949), 297-304. https://doi.org/10.2307/1969454