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High-entropy dual functions over finite fields and locally decodable codes

Part of: Sequences and sets

Published online by Cambridge University Press:  08 March 2021

Jop Briët  and
Farrokh Labib
Jop Briët
Affiliation:
CWI, Science Park 123, 1098 XGAmsterdam, The Netherlands; E-mails: j.briet@cwi.nl and labib@cwi.nl.
Farrokh Labib
Affiliation:
CWI, Science Park 123, 1098 XGAmsterdam, The Netherlands; E-mails: j.briet@cwi.nl and labib@cwi.nl.

Abstract

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We show that for infinitely many primes p there exist dual functions of order k over ${\mathbb{F}}_p^n$ that cannot be approximated in $L_\infty $-distance by polynomial phase functions of degree $k-1$. This answers in the negative a natural finite-field analogue of a problem of Frantzikinakis on $L_\infty $-approximations of dual functions over ${\mathbb{N}}$ (a.k.a. multiple correlation sequences) by nilsequences.

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity
Secondary: 11B25: Arithmetic progressions
Type
Analysis
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e19
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Altman, D., ‘On Szemerédi’s theorem with differences from a random set’, Acta Arith. 195 (2020), 97–108.CrossRefGoogle Scholar
Arora, S., Lund, C., Motwani, R., Sudan, M., and Szegedy, M., ‘Proof verification and the hardness of approximation problems’, J. ACM. 45(3) (1998), 501–555.CrossRefGoogle Scholar
Arora, S. and Safra, S., ‘Probabilistic checking of proofs: a new characterization of NP’, J. ACM 45(1) (1998), 70–122.CrossRefGoogle Scholar
Blum, M. and Kannan, S., ‘Designing programs that check their work’, J. ACM 42(1) (1995), 269–291.CrossRefGoogle Scholar
Briët, J., ‘Subspaces of tensors with high analytic rank’, Online J. Anal. Comb. 2020, arXiv:1908.04169.Google Scholar
Briët, J., Dvir, Z., and Gopi, S., ‘Outlaw distributions and locally decodable codes’, Theory Comput. 15(12) (2019), 1–24.CrossRefGoogle Scholar
Briët, J. and Gopi, S., ‘Gaussian width bounds with applications to arithmetic progressions in random settings’, Int. Math. Res. Not. 2020(22) November 2020, 8673–8696.Google Scholar
Briët, J. and Green, B., ‘Multiple correlation sequences not approximable by nilsequences, 2020, preprint arXiv:2010.14960.Google Scholar
Briët, J., Naor, A., and Regev, O., ‘Locally decodable codes and the failure of cotype for projective tensor products’, Electron. Res. Announc. Math. Sci. 19 (2012), 120–130.Google Scholar
Chor, B., Goldreich, O., Kushilevitz, E., and Sudan, M., ‘Private information retrieval’, J. ACM 45(6) (1998), 965–982.CrossRefGoogle Scholar
Efremenko, K., ‘3-Query locally decodable codes of subexponential length’, SIAM J. Comput. 41(6) (2012), 1694–1703.CrossRefGoogle Scholar
Frantzikinakis, N. ‘Some open problems on multiple ergodic averages’, Bull. Hellenic Math. Soc. 60 (2016), 41–90.Google Scholar
Frantzikinakis, N., Lesigne, E., and Wierdl, M., ‘Random sequences and pointwise convergence of multiple ergodic averages’, Indiana Univ. Math. J., 61(2) (2012), 585–617.CrossRefGoogle Scholar
Frantzikinakis, N., Lesigne, E., and Wierdl, M., ‘Random differences in Szemerédi’s theorem and related results’, J. Anal. Math. 130 (2016), 91–133.CrossRefGoogle Scholar
Furstenberg, H. and Katznelson, Y., ‘A density version of the Hales-Jewett theorem’, J. Anal. Math. 57(1) (1991), 64–119.CrossRefGoogle Scholar
GIMPS. Great Internet Mersenne prime search, December 21, 2018. URL: https://www.mersenne.org/.Google Scholar
Gopalan, P., Huang, C., Simitci, H., and Yekhanin, S., ‘On the locality of codeword symbols’, IEEE Trans. Inform. Theory 58(11) (2012), 6925–6934.CrossRefGoogle Scholar
Gopi, S., Locality in Coding Theory, PhD Thesis (Princeton University, 2018).Google Scholar
Gowers, W. T., ‘Decompositions, approximate structure, transference, and the Hahn–Banach theorem’, Bull. Lon. Math. Soc. 42(4) (2010), 573–606.CrossRefGoogle Scholar
Green, B., Tao, T., and Ziegler, T., ‘An inverse theorem for the Gowers ${U}^{s+1}\left[N\right]$-norm’, Ann. Math. Second Series, 176(2) (2012), 1231–1372.CrossRefGoogle Scholar
Katz, J. and Trevisan, L., ‘On the efficiency of local decoding procedures for error-correcting codes’, in Proc. 32nd STOC (ACM Press, New York, NY, United States 2000), 80–86.Google Scholar
Kerenidis, I. and de Wolf, R., ‘Exponential lower bound for 2-query locally decodable codes via a quantum argument’, J. Comput. System Sci. 69(3) (2004), 395–420.CrossRefGoogle Scholar
Lidl, R. and Niederreiter, H., Finite Fields, Vol. 20 (Cambridge University Press, The Edinburgh Building, Cambridge CB2 8RU, UK 1997, Published in New York, USA).Google Scholar
Milman, V. D. and Schechtman, G., Asymptotic Theory of Finite-Dimensional Normed Spaces, Vol. 1200 of Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1986). With an appendix by M. Gromov.Google Scholar
Raghavendra, P., ‘A note on Yekhanin’s locally decodable codes’, Electron. Coll. Comput. Complex. 14(16) (2007), 1–8.Google Scholar
Stewart, C. L., ‘On divisors of Lucas and Lehmer numbers’, Acta Math. 211(2) (2013), 291–314.CrossRefGoogle Scholar
Tao, T. and Ziegler, T., ‘The inverse conjecture for the Gowers norm over finite fields via the correspondence principle’, Anal. PDE 3(1) (2010), 1–20.CrossRefGoogle Scholar
Tao, T. and Ziegler, T., ‘The inverse conjecture for the Gowers norm over finite fields in low characteristic’, Ann. Comb. 16(1) (2012), 121–188.CrossRefGoogle Scholar
Woodruff, D., ‘New lower bounds for general locally decodable codes’, Electron. Coll. Comput. Complex. 14(6) (2007), 1–19.Google Scholar
Yekhanin, S., ‘Towards 3-query locally decodable codes of subexponential length’, J. ACM 55(1) (2008), 1:1–1:16.CrossRefGoogle Scholar
Yekhanin, S., ‘Locally decodable codes’, Found. Trends Theor. Comput. Sci. 6(3) (2012), 139–255.CrossRefGoogle Scholar