thông tin chi tiết - Computational Complexity - # Corner-Free Sets

Improved Upper Bounds on the Density of Corner-Free Sets in Quasirandom Groups and Applications to Communication Complexity

Q: Can the techniques used in this paper be extended to analyze higher-dimensional analogs of corner-free sets, and what implications might such extensions have for communication complexity in NOF models with more players?

Extending the techniques to higher dimensions presents exciting possibilities and significant challenges. Here's a breakdown: Potential Extensions: Higher-Dimensional Corners: The concept of a corner naturally generalizes. In $\mathbb{F}_2^n$, a $d$-dimensional corner is a set of $d+1$ points where one point is obtained by taking the coordinate-wise sum of a subset of the other $d$ points. Generalized Permutation Functions: The notion of a permutation function can be extended to higher dimensions. A $d$-dimensional permutation function would be a subset of $(\mathbb{F}_2^n)^{d+1}$ where fixing any $d$ coordinates uniquely determines the remaining one. Grid Norms: Grid norms, crucial for capturing rectangular structure, can be defined for functions on higher-dimensional domains. Challenges and Implications: Pseudorandomness: Defining and establishing appropriate pseudorandomness notions for higher-dimensional objects (like hypercubes) become more intricate. The current proof relies heavily on the properties of rectangles and their generalizations. Analytic Techniques: The analysis using grid norms and Hölder's inequality might need substantial adaptations for higher-order structures. New inequalities or techniques might be required to control the relevant terms effectively. NOF Communication Complexity: Success in extending these techniques could lead to lower bounds for NOF communication complexity of functions with more than three players. The connection between corner-free sets and Exactly-N hints at a potential pathway for such results. In summary, while extending the techniques to higher dimensions is not straightforward, it holds promise for advancing our understanding of both additive combinatorics and communication complexity.

Q: Could there be alternative approaches, perhaps based on different pseudorandomness notions or analytical tools, that might yield even tighter bounds on the density of corner-free sets in quasirandom groups?

Yes, alternative approaches could potentially lead to tighter bounds. Here are some avenues to explore: Stronger Pseudorandomness: The current notion of pseudorandomness against cubes might be strengthened. For instance, one could require that the set behaves pseudorandomly even when restricted to specific families of highly structured subsets, not just cubes. Spectral Methods: Instead of grid norms, tools from spectral graph theory, such as eigenvalues of Cayley graphs associated with the quasirandom group, might provide different insights into the structure of corner-free sets. Combinatorial Regularity: Variants of Szemerédi's regularity lemma, tailored for quasirandom groups, could potentially be employed. However, regularity-based approaches often lead to weak quantitative bounds. Structure vs. Randomness Dichotomy: A promising direction is to explore a "structure vs. randomness" dichotomy for corner-free sets in quasirandom groups. This would involve showing that any such set is either close to a structured object (whose size can be bounded) or exhibits random-like behavior (which would also limit its density). Achieving optimal bounds, such as showing that corner-free sets in SL2(Fp) have density at most |G|^(2-ε) for some ε > 0, might require a combination of novel ideas and a deeper understanding of the interplay between the group structure and the forbidden configuration.

Q: What are the potential implications of these findings for other areas of mathematics or computer science where the structure of sets with forbidden patterns plays a crucial role, such as coding theory or Ramsey theory?

The findings in this paper have the potential to impact several areas where the avoidance of specific patterns is crucial: Coding Theory: List Decoding: Corner-free sets can be viewed as codes with certain distance properties. The techniques developed might inspire new constructions or analyses of codes with good list-decoding capabilities. Locally Decodable Codes: The pseudorandomness properties and the connections to communication complexity could be relevant for designing or analyzing locally decodable codes, where one aims to retrieve specific bits of information from a corrupted codeword by querying only a few positions. Ramsey Theory: Quantitative Bounds: The improved bounds on corner-free sets might lead to better quantitative bounds in related Ramsey-type problems, where one seeks the minimum size of a structure that guarantees the existence of certain monochromatic substructures. New Constructions: The methods used to construct or analyze corner-free sets could inspire new constructions of sets avoiding other patterns relevant to Ramsey theory. Theoretical Computer Science: Property Testing: The ideas of pseudorandomness and structure vs. randomness could find applications in property testing, where one aims to design efficient algorithms to determine if a large object satisfies a given property or is far from satisfying it. Complexity Theory: The connections to communication complexity suggest potential links to other areas of complexity theory, such as circuit complexity or proof complexity. Overall, the study of sets with forbidden patterns, like corner-free sets, is a fundamental theme across mathematics and computer science. The techniques and insights from this paper could contribute to a deeper understanding of these structures and their applications in various domains.

Khái niệm cốt lõi

This paper presents a novel combinatorial theorem that yields improved upper bounds on the density of corner-free sets within quasirandom groups, leading to advancements in understanding the communication complexity of specific functions in the Number-on-Forehead model.

Tóm tắt

Bibliographic Information: Jaber, M., Lovett, S., & Ostuni, A. (2024). Corners in Quasirandom Groups via Sparse Mixing. arXiv preprint arXiv:2411.02702v1.
Research Objective: This paper aims to improve the upper bounds on the density of corner-free sets in quasirandom groups and apply these findings to enhance the lower bounds on the communication complexity of certain functions in the 3-player Number-on-Forehead (NOF) model.
Methodology: The authors develop a general combinatorial theorem that leverages the concept of "sparse mixing" to analyze the structure of corner-free sets within quasirandom groups. This theorem extends previous work on three-term arithmetic progressions and employs grid norms to measure rectangular structure within sets.
Key Findings: The paper demonstrates that corner-free subsets of specific quasirandom groups, such as SL2(Fp), have a density bounded by a quasi-polynomial function of the group size. This significantly improves upon the previous inverse poly-logarithmic bounds. Consequently, the communication complexity of the Exactly-N function over these groups in the 3-player NOF model is shown to be at least logarithmic to the base 1/4 of the group size.
Main Conclusions: The research provides a significant advancement in understanding the limitations of corner-free sets in quasirandom groups, with implications for communication complexity. The use of sparse mixing and grid norms offers a new approach to tackling problems in additive combinatorics.
Significance: This work contributes significantly to both additive combinatorics and communication complexity. The improved bounds on corner-free sets have implications for other related problems, while the communication complexity results shed light on the limitations of the NOF model.
Limitations and Future Research: The authors acknowledge that their results primarily apply to sparse sets and suggest exploring whether these techniques can be generalized to denser sets. Further research could investigate extending these bounds to corner-free sets over integers or finite fields, potentially leading to optimal separations between randomized and deterministic 3-NOF protocols. Additionally, extending the NOF lower bounds to more than three players is an open avenue for future work.

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Thống kê

Any non-deterministic 3-NOF protocol computing Exactly-N over G = SL2(Fp) for prime p requires Ω(log^(1/4) |G|) bits of communication.
Corner-free subsets of G = SL2(Fp) for prime p have size at most δ|G|^2 for δ = exp(-Ω(log^(1/4) |G|)).

Trích dẫn

"We improve the best known upper bounds on the density of corner-free sets over quasirandom groups from inverse poly-logarithmic to quasi-polynomial."
"We make similarly substantial improvements to the best known lower bounds on the communication complexity of a large class of permutation functions in the 3-player Number-on-Forehead model."
"Underpinning both results is a general combinatorial theorem that extends the recent work of Kelley, Lovett, and Meka (STOC’24), itself a development of ideas from the breakthrough result of Kelley and Meka on three-term arithmetic progressions (FOCS’23)."

Thông tin chi tiết chính được chắt lọc từ

Corners in Quasirandom Groups via Sparse Mixing

by Michael Jabe... lúc arxiv.org 11-06-2024

https://arxiv.org/pdf/2411.02702.pdf

Corners in Quasirandom Groups via Sparse Mixing

Yêu cầu sâu hơn

Can the techniques used in this paper be extended to analyze higher-dimensional analogs of corner-free sets, and what implications might such extensions have for communication complexity in NOF models with more players?

Extending the techniques to higher dimensions presents exciting possibilities and significant challenges. Here's a breakdown:
Potential Extensions:

Higher-Dimensional Corners: The concept of a corner naturally generalizes. In $\mathbb{F}_2^n$, a $d$-dimensional corner is a set of $d+1$ points where one point is obtained by taking the coordinate-wise sum of a subset of the other $d$ points.
Generalized Permutation Functions:  The notion of a permutation function can be extended to higher dimensions. A $d$-dimensional permutation function would be a subset of $(\mathbb{F}_2^n)^{d+1}$ where fixing any $d$ coordinates uniquely determines the remaining one.
Grid Norms: Grid norms, crucial for capturing rectangular structure, can be defined for functions on higher-dimensional domains.
Challenges and Implications:

Pseudorandomness:  Defining and establishing appropriate pseudorandomness notions for higher-dimensional objects (like hypercubes) become more intricate. The current proof relies heavily on the properties of rectangles and their generalizations.
Analytic Techniques:  The analysis using grid norms and Hölder's inequality might need substantial adaptations for higher-order structures. New inequalities or techniques might be required to control the relevant terms effectively.
NOF Communication Complexity:  Success in extending these techniques could lead to lower bounds for NOF communication complexity of functions with more than three players.  The connection between corner-free sets and Exactly-N hints at a potential pathway for such results.
In summary, while extending the techniques to higher dimensions is not straightforward, it holds promise for advancing our understanding of both additive combinatorics and communication complexity.

Could there be alternative approaches, perhaps based on different pseudorandomness notions or analytical tools, that might yield even tighter bounds on the density of corner-free sets in quasirandom groups?

Yes, alternative approaches could potentially lead to tighter bounds. Here are some avenues to explore:

Stronger Pseudorandomness: The current notion of pseudorandomness against cubes might be strengthened. For instance, one could require that the set behaves pseudorandomly even when restricted to specific families of highly structured subsets, not just cubes.
Spectral Methods:  Instead of grid norms, tools from spectral graph theory, such as eigenvalues of Cayley graphs associated with the quasirandom group, might provide different insights into the structure of corner-free sets.
Combinatorial Regularity:  Variants of Szemerédi's regularity lemma, tailored for quasirandom groups, could potentially be employed. However, regularity-based approaches often lead to weak quantitative bounds.
Structure vs. Randomness Dichotomy:  A promising direction is to explore a "structure vs. randomness" dichotomy for corner-free sets in quasirandom groups. This would involve showing that any such set is either close to a structured object (whose size can be bounded) or exhibits random-like behavior (which would also limit its density).
Achieving optimal bounds, such as showing that corner-free sets in SL2(Fp) have density at most |G|^(2-ε) for some ε > 0, might require a combination of novel ideas and a deeper understanding of the interplay between the group structure and the forbidden configuration.

What are the potential implications of these findings for other areas of mathematics or computer science where the structure of sets with forbidden patterns plays a crucial role, such as coding theory or Ramsey theory?

The findings in this paper have the potential to impact several areas where the avoidance of specific patterns is crucial:

Coding Theory:

List Decoding: Corner-free sets can be viewed as codes with certain distance properties. The techniques developed might inspire new constructions or analyses of codes with good list-decoding capabilities.
Locally Decodable Codes:  The pseudorandomness properties and the connections to communication complexity could be relevant for designing or analyzing locally decodable codes, where one aims to retrieve specific bits of information from a corrupted codeword by querying only a few positions.

Ramsey Theory:

Quantitative Bounds: The improved bounds on corner-free sets might lead to better quantitative bounds in related Ramsey-type problems, where one seeks the minimum size of a structure that guarantees the existence of certain monochromatic substructures.
New Constructions: The methods used to construct or analyze corner-free sets could inspire new constructions of sets avoiding other patterns relevant to Ramsey theory.

Theoretical Computer Science:

Property Testing: The ideas of pseudorandomness and structure vs. randomness could find applications in property testing, where one aims to design efficient algorithms to determine if a large object satisfies a given property or is far from satisfying it.
Complexity Theory: The connections to communication complexity suggest potential links to other areas of complexity theory, such as circuit complexity or proof complexity.
Overall, the study of sets with forbidden patterns, like corner-free sets, is a fundamental theme across mathematics and computer science. The techniques and insights from this paper could contribute to a deeper understanding of these structures and their applications in various domains.