Gaussian width bounds with applications to arithmetic progressions in random settings

Motivated by two problems on arithmetic progressions (APs)---concerning large deviations for AP counts in random sets and random differences in Szemer\'edi's theorem---we prove upper bounds on the Gaussian width of the image of the $n$-dimensional Boolean hypercube under a mapping $\psi:\mathbb{R}^n\to\mathbb{R}^k$, where each coordinate is a constant-degree multilinear polynomial with $0/1$ coefficients. We show the following applications of our bounds. Let $[\mathbb{Z}/N\mathbb{Z}]_p$ be the random subset of $\mathbb{Z}/N\mathbb{Z}$ containing each element independently with probability $p$. - Let $X_k$ be the number of $k$-term APs in $[\mathbb{Z}/N\mathbb{Z}]_p$. We show that a precise estimate on the large deviation rate $\log\Pr[X_k \geq (1+\delta)\mathbb{E}X_k]$ due to Bhattacharya, Ganguly, Shao and Zhao is valid if $p \geq \omega(N^{-c_k}\log N)$ for $c_k = (6k\lceil(k-1)/2\rceil)^{-1}$, which slightly improves their bound of $c_k = (6k(k-1))^{-1}$ for $k \geq 5$ (and matching their $c_3$ and $c_4$). - A set $D\subseteq \mathbb{Z}/N\mathbb{Z}$ is $\ell$-intersective if every dense subset of $\mathbb{Z}/N\mathbb{Z}$ contains a non-trivial $(\ell+1)$-term AP with common difference in $D$. We show that $[\mathbb{Z}/N\mathbb{Z}]_p$ is $\ell$-intersective with probability $1 - o_N(1)$ provided $p \geq \omega(N^{-\beta_\ell}\log N)$ for $\beta_\ell = (\lceil(\ell+1)/2\rceil)^{-1}$, improving the bound $\beta_\ell =(\ell+1)^{-1}$ due to Frantzikinakis, Lesigne and Wierdl for $\ell \geq 2$ and reproving more directly the same result shown recently by the authors and Dvir. In addition, we discuss some intriguing connections with special kinds of error correcting codes (locally decodable codes) and the Banach-space notion of type for injective tensor products of $\ell_p$-spaces.

[1]  Nikos Frantzikinakis,et al.  Random Sequences and Pointwise Convergence of Multiple Ergodic Averages , 2010, 1012.1130.

[2]  N. Tomczak-Jaegermann.  The moduli of smoothness and convexity and the Rademacher averages of the trace classes $S_{p}$ (1≤p<∞) , 1974 .

[3]  Eli Upfal,et al.  Probability and Computing: Randomized Algorithms and Probabilistic Analysis , 2005 .

[4]  Trevor D. Wooley,et al.  Multiple recurrence and convergence along the primes , 2010, 1001.4081.

[5]  Vitaly Bergelson,et al.  Polynomial extensions of van der Waerden’s and Szemerédi’s theorems , 1996 .

[6]  Luca Trevisan,et al.  Lower bounds for linear locally decodable codes and private information retrieval , 2006, computational complexity.

[7]  Yufei Zhao,et al.  On the variational problem for upper tails in sparse random graphs , 2014, Random Struct. Algorithms.

[8]  Zeev Dvir,et al.  Outlaw Distributions and Locally Decodable Codes , 2016, ITCS.

[9]  Nikos Frantzikinakis,et al.  Random differences in Szemerédi’s theorem and related results , 2016 .

[10]  R. Ryan.  Introduction to Tensor Products of Banach Spaces , 2002 .

[11]  Ronen Eldan,et al.  Gaussian-width gradient complexity, reverse log-Sobolev inequalities and nonlinear large deviations , 2016, Geometric and Functional Analysis.

[12]  P. Varnavides,et al.  On Certain Sets of Positive Density , 1959 .

[13]  L. Warnke.  Upper tails for arithmetic progressions in random subsets , 2016, 1612.08559.

[14]  Bhaswar B. Bhattacharya,et al.  Upper tails and independence polynomials in random graphs , 2015, 1507.04074.

[15]  A. Sárközy,et al.  On difference sets of sequences of integers. III , 1978 .

[16]  Sergey Yekhanin,et al.  Locally Decodable Codes , 2012, Found. Trends Theor. Comput. Sci..

[17]  Jop Briët,et al.  On Embeddings of l1k from Locally Decodable Codes , 2016, Electron. Colloquium Comput. Complex..

[18]  M. Talagrand.  Upper and Lower Bounds for Stochastic Processes , 2021, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics.

[19]  Amir Dembo,et al.  Nonlinear large deviations , 2014, 1401.3495.

[20]  Ronald de Wolf,et al.  Exponential lower bound for 2-query locally decodable codes via a quantum argument , 2002, STOC '03.

[21]  Luca Trevisan,et al.  Lower bounds for linear locally decodable codes and private information retrieval , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[22]  A. SARICOZY.  ON DIFFERENCE SETS OF SEQUENCES OF INTEGERS . III , 1978 .

[23]  Jonathan Katz,et al.  On the efficiency of local decoding procedures for error-correcting codes , 2000, STOC '00.

[24]  E. Szemerédi.  On sets of integers containing k elements in arithmetic progression , 1975 .

[25]  Assaf Naor,et al.  Locally decodable codes and the failure of cotype for projective tensor products , 2012, ArXiv.

[26]  Michael Christ,et al.  On random multilinear operator inequalities , 2011, 1108.5655.

[27]  T. Tao.  The ergodic and combinatorial approaches to Szemerédi's theorem , 2006, math/0604456.