Revisiting the Sanders-Freiman-Ruzsa Theorem in Fpn\mathbb{F}_p^n and its Application to Non-malleable Codes

Non-malleable codes (NMCs) protect sensitive data against degrees of corruption that prohibit error detection, ensuring instead that a corrupted codeword decodes correctly or to something that bears little relation to the original message. The split-state model, in which codewords consist of two blocks, considers adversaries who tamper with either block arbitrarily but independently of the other. The simplest construction in this model, due to Aggarwal, Dodis, and Lovett (STOC'14), was shown to give NMCs sending k-bit messages to $O(k^7)$-bit codewords. It is conjectured, however, that the construction allows linear-length codewords. Towards resolving this conjecture, we show that the construction allows for code-length $O(k^5)$. This is achieved by analysing a special case of Sanders's Bogolyubov-Ruzsa theorem for general Abelian groups. Closely following the excellent exposition of this result for the group $\mathbb{F}_2^n$ by Lovett, we expose its dependence on $p$ for the group $\mathbb{F}_p^n$, where $p$ is a prime

Gaussian width bounds with applications to arithmetic progressions in random settings

Motivated by problems on random differences in Szemer\'{e}di's theorem and on large deviations for arithmetic progressions in random sets, we prove upper bounds on the Gaussian width of point sets that are formed by 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 $p$. $\bullet$ A set $D\subseteq \mathbb{Z}/N\mathbb{Z}$ is $\ell$-intersective if any dense subset of $\mathbb{Z}/N\mathbb{Z}$ contains a proper $(\ell+1)$-term arithmetic progression with common difference in $D$. Our main result implies that $[\mathbb{Z}/N\mathbb{Z}]_p$ is $\ell$-intersective with probability $1 - o(1)$ provided $p \geq \omega(N^{-\beta_\ell}\log N)$ for $\beta_\ell = (\lceil(\ell+1)/2\rceil)^{-1}$. This gives a polynomial improvement for all $\ell \ge 3$ of a previous bound due to Frantzikinakis, Lesigne and Wierdl, and reproves more directly the same improvement shown recently by the authors and Dvir. $\bullet$ Let $X_k$ be the number of $k$-term arithmetic progressions in $[\mathbb{Z}/N\mathbb{Z}]_p$ and consider the large deviation rate $\rho_k(\delta) = \log\Pr[X_k \geq (1+\delta)\mathbb{E}X_k]$. We give quadratic improvements of the best-known range of $p$ for which a highly precise estimate of $\rho_k(\delta)$ due to Bhattacharya, Ganguly, Shao and Zhao is valid for all odd $k \geq 5$. We also discuss connections with error correcting codes (locally decodable codes) and the Banach-space notion of type for injective tensor products of $\ell_p$-spaces.Comment: 18 pages, some typos fixe

On the orthogonal rank of Cayley graphs and impossibility of quantum round elimination

After Bob sends Alice a bit, she responds with a lengthy reply. At the cost of a factor of two in the total communication, Alice could just as well have given the two possible replies without listening and have Bob select which applies to him. Motivated by a conjecture stating that this form of "round elimination" is impossible in exact quantum communication complexity, we study the orthogonal rank and a symmetric variant thereof for a certain family of Cayley graphs. The orthogonal rank of a graph is the smallest number $d$ for which one can label each vertex with a nonzero $d$-dimensional complex vector such that adjacent vertices receive orthogonal vectors. We show an exp$(n)$ lower bound on the orthogonal rank of the graph on $\{0,1\}^n$ in which two strings are adjacent if they have Hamming distance at least $n/2$. In combination with previous work, this implies an affirmative answer to the above conjecture.Comment: 13 page

Failure of the trilinear operator space Grothendieck theorem

We give a counterexample to a trilinear version of the operator space Grothendieck theorem. In particular, we show that for trilinear forms on $\ell_\infty$, the ratio of the symmetrized completely bounded norm and the jointly completely bounded norm is in general unbounded, answering a question of Pisier. The proof is based on a non-commutative version of the generalized von Neumann inequality from additive combinatorics.Comment: Reformatted for Discrete Analysi

Erosion dynamics of a wet granular medium

Liquid may give strong cohesion properties to a granular medium, and confer a solid-like behavior. We study the erosion of a fixed circular aggregate of wet granular matter subjected to a flow of dry grains inside a half-filled rotating drum. During the rotation, the dry grains flow around the fixed obstacle. We show that its diameter decreases linearly with time for low liquid content, as wet grains are pulled-out of the aggregate. This erosion phenomenon is governed by the properties of the liquids. The erosion rate decreases exponentially with the surface tension while it depends on the viscosity to the power -1. We propose a model based on the force fluctuations arising inside the flow, explaining both dependencies: the capillary force acts as a threshold and the viscosity controls the erosion time scale. We also provide experiments using different flowing grains confirming our model

Locally Decodable Quantum Codes

We study a quantum analogue of locally decodable error-correcting codes. A q-query locally decodable quantum code encodes n classical bits in an m-qubit state, in such a way that each of the encoded bits can be recovered with high probability by a measurement on at most q qubits of the quantum code, even if a constant fraction of its qubits have been corrupted adversarially. We show that such a quantum code can be transformed into a classical q-query locally decodable code of the same length that can be decoded well on average (albeit with smaller success probability and noise-tolerance). This shows, roughly speaking, that q-query quantum codes are not significantly better than q-query classical codes, at least for constant or small q.Comment: 15 pages, LaTe

Locally decodable codes and the failure of cotype for projective tensor products

It is shown that for every $p\in (1,\infty)$ there exists a Banach space $X$ of finite cotype such that the projective tensor product \ell_p\tp X fails to have finite cotype. More generally, if $p_1,p_2,p_3\in (1,\infty)$ satisfy $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\le 1$ then \ell_{p_1}\tp\ell_{p_2}\tp\ell_{p_3} does not have finite cotype. This is a proved via a connection to the theory of locally decodable codes

Crucial role of side walls for granular surface flows: consequences for the rheology

In this paper we study the steady uniform flows that develop when granular material is released from a hopper on top of a static pile in a channel. We more specifically focus on the role of side walls by carrying out experiments in setup of different widths, from narrow channels 20 particle diameters wide to channels 600 particle diameters wide. Results show that steady flows on pile are entirely controlled by side wall effects. A theoretical model, taking into account the wall friction and based on a simple local constitutive law recently proposed for other granular flow configurations (GDR MiDi 2004), gives predictions in quantitative agreement with the measurements. This result gives new insights in our understanding of free surface granular flows and strongly supports the relevance of the constitutive law proposed.Comment: a forgotten square root in Appendix B (Eq B4), and corrected coefficients in Appendix C; 25 pages, 17 figures, published in J. Fluid Mec

A generalized Grothendieck inequality and entanglement in XOR games

Suppose Alice and Bob make local two-outcome measurements on a shared entangled state. For any d, we show that there are correlations that can only be reproduced if the local dimension is at least d. This resolves a conjecture of Brunner et al. Phys. Rev. Lett. 100, 210503 (2008) and establishes that the amount of entanglement required to maximally violate a Bell inequality must depend on the number of measurement settings, not just the number of measurement outcomes. We prove this result by establishing the first lower bounds on a new generalization of Grothendieck's constant.Comment: Version submitted to QIP on 10-20-08. See also arxiv:0812.1572 for related results, obtained independentl