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# Tight Chang’s-Lemma-Type Bounds for Boolean Functions

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## Cite As

Sourav Chakraborty, Nikhil S. Mande, Rajat Mittal, Tulasimohan Molli, Manaswi Paraashar, and Swagato Sanyal. Tight Chang’s-Lemma-Type Bounds for Boolean Functions. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 10:1-10:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.FSTTCS.2021.10

## Abstract

Chang’s lemma (Duke Mathematical Journal, 2002) is a classical result in mathematics, with applications spanning across additive combinatorics, combinatorial number theory, analysis of Boolean functions, communication complexity and algorithm design. For a Boolean function f that takes values in {-1, 1} let r(f) denote its Fourier rank (i.e., the dimension of the span of its Fourier support). For each positive threshold t, Chang’s lemma provides a lower bound on δ(f) := Pr[f(x) = -1] in terms of the dimension of the span of its characters with Fourier coefficients of magnitude at least 1/t. In this work we examine the tightness of Chang’s lemma with respect to the following three natural settings of the threshold: - the Fourier sparsity of f, denoted k(f), - the Fourier max-supp-entropy of f, denoted k'(f), defined to be the maximum value of the reciprocal of the absolute value of a non-zero Fourier coefficient, - the Fourier max-rank-entropy of f, denoted k''(f), defined to be the minimum t such that characters whose coefficients are at least 1/t in magnitude span a r(f)-dimensional space. In this work we prove new lower bounds on δ(f) in terms of the above measures. One of our lower bounds, δ(f) = Ω(r(f)²/(k(f) log² k(f))), subsumes and refines the previously best known upper bound r(f) = O(√{k(f)}log k(f)) on r(f) in terms of k(f) by Sanyal (Theory of Computing, 2019). We improve upon this bound and show r(f) = O(√{k(f)δ(f)}log k(f)). Another lower bound, δ(f) = Ω(r(f)/(k''(f) log k(f))), is based on our improvement of a bound by Chattopadhyay, Hatami, Lovett and Tal (ITCS, 2019) on the sum of absolute values of level-1 Fourier coefficients in terms of 𝔽₂-degree. We further show that Chang’s lemma for the above-mentioned choices of the threshold is asymptotically outperformed by our bounds for most settings of the parameters involved. Next, we show that our bounds are tight for a wide range of the parameters involved, by constructing functions witnessing their tightness. All the functions we construct are modifications of the Addressing function, where we replace certain input variables by suitable functions. Our final contribution is to construct Boolean functions f for which our lower bounds asymptotically match δ(f), and for any choice of the threshold t, the lower bound obtained from Chang’s lemma is asymptotically smaller than δ(f). Our results imply more refined deterministic one-way communication complexity upper bounds for XOR functions. Given the wide-ranging application of Chang’s lemma to areas like additive combinatorics, learning theory and communication complexity, we strongly feel that our refinements of Chang’s lemma will find many more applications.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Oracles and decision trees
• Theory of computation → Communication complexity
##### Keywords
• Analysis of Boolean functions
• Chang’s lemma
• Parity decision trees
• Fourier dimension

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## References

1. Srinivasan Arunachalam, Sourav Chakraborty, Troy Lee, Manaswi Paraashar, and Ronald de Wolf. Two new results about quantum exact learning. In 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, volume 132 of LIPIcs, pages 16:1-16:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019.
2. Eli Ben-Sasson, Noga Ron-Zewi, Madhur Tulsiani, and Julia Wolf. Sampling-based proofs of almost-periodicity results and algorithmic applications. In 41st International Colloquium on Automata, Languages, and Programming ICALP 2014, volume 8572 of Lecture Notes in Computer Science, pages 955-966. Springer, 2014.
3. Anna Bernasconi and Bruno Codenotti. Spectral analysis of Boolean functions as a graph eigenvalue problem. IEEE Trans. Computers, 48(3):345-351, 1999. URL: https://doi.org/10.1109/12.755000.
4. Thomas F Bloom. A quantitative improvement for Roth’s theorem on arithmetic progressions. Journal of the London Mathematical Society, 93(3):643-663, 2016.
5. Thomas F Bloom and Olof Sisask. Breaking the logarithmic barrier in Roth’s theorem on arithmetic progressions. arXiv preprint, 2020. URL: http://arxiv.org/abs/2007.03528.
6. Sourav Chakraborty, Nikhil S. Mande, Rajat Mittal, Tulasimohan Molli, Manaswi Paraashar, and Swagato Sanyal. Tight chang’s-lemma-type bounds for boolean functions. CoRR, abs/2012.02335, 2020. URL: http://arxiv.org/abs/2012.02335.
7. Siu On Chan, James R. Lee, Prasad Raghavendra, and David Steurer. Approximate constraint satisfaction requires large LP relaxations. J. ACM, 63(4):34:1-34:22, 2016.
8. Mei-Chu Chang. A polynomial bound in Freiman’s theorem. Duke mathematical journal, 113(3):399-419, 2002.
9. Eshan Chattopadhyay, Pooya Hatami, Shachar Lovett, and Avishay Tal. Pseudorandom generators from the second Fourier level and applications to AC0 with parity gates. In 10th Innovations in Theoretical Computer Science Conference, ITCS, pages 22:1-22:15, 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.22.
10. Ehud Friedgut, Jeff Kahn, Gil Kalai, and Nathan Keller. Chvátal’s conjecture and correlation inequalities. J. Comb. Theory, Ser. A, 156:22-43, 2018.
11. Parikshit Gopalan, Ryan O'Donnell, Rocco A. Servedio, Amir Shpilka, and Karl Wimmer. Testing Fourier dimensionality and sparsity. SIAM J. Comput., 40(4):1075-1100, 2011. URL: https://doi.org/10.1137/100785429.
12. Ben Green. Arithmetic progressions in sumsets. Geometric and Functional Analysis GAFA, 12(3):584-597, 2002.
13. Ben Green. Some constructions in the inverse spectral theory of cyclic groups. Combinatorics, Probability and Computing, 12(2):127-138, 2003.
14. Ben Green. Spectral structure of sets of integers. In Fourier analysis and convexity, pages 83-96. Springer, 2004.
15. Ben Green and Imre Z. Ruzsa. Freiman’s theorem in an arbitrary abelian group. Journal of the London Mathematical Society, 75(1):163-175, 2007.
16. Ben Green and Tom Sanders. Boolean functions with small spectral norm. Geometric and Functional Analysis GAFA, 18:144-162, 2008.
17. Tom Gur and Omer Tamuz. Testing Booleanity and the uncertainty principle. Chic. J. Theor. Comput. Sci., 2013, 2013.
18. Pooya Hatami, Raghav Kulkarni, and Denis Pankratov. Variations on the sensitivity conjecture. Theory of Computing, 4:1-27, 2011. URL: https://doi.org/10.4086/toc.gs.2011.004.
19. Kaave Hosseini, Shachar Lovett, and Grigory Yaroslavtsev. Optimality of linear sketching under modular updates. In 34th Computational Complexity Conference, CCC 2019, volume 137 of LIPIcs, pages 13:1-13:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019.
20. Russell Impagliazzo, Cristopher Moore, and Alexander Russell. An entropic proof of Chang’s inequality. SIAM J. Discret. Math., 28(1):173-176, 2014.
21. Shachar Lovett. Communication is bounded by root of rank. Journal of the ACM (JACM), 63(1):1-9, 2016.
22. Ashley Montanaro and Tobias Osborne. On the communication complexity of XOR functions. CoRR, abs/0909.3392, 2009. URL: http://arxiv.org/abs/0909.3392.
23. Tom Sanders. Additive structures in sumsets. Mathematical Proceedings of the Cambridge Philosophical Society, 144(2):289-316, 2008.
24. Tom Sanders. On Roth’s theorem on progressions. Annals of Mathematics, 174:619-636, 2011.
25. Swagato Sanyal. Fourier sparsity and dimension. Theory of Computing, 15(11):1-13, 2019. URL: https://doi.org/10.4086/toc.2019.v015a011.
26. Hing Yin Tsang, Chung Hoi Wong, Ning Xie, and Shengyu Zhang. Fourier sparsity, spectral norm, and the log-rank conjecture. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 658-667, 2013. URL: https://doi.org/10.1109/FOCS.2013.76.