Discrete Cube Research Articles

Hypercontractivity for global functions and sharp thresholds

The classical hypercontractive inequality for the noise operator on the discrete cube plays a crucial role in many of the fundamental results in the Analysis of Boolean functions, such as the Kahn-Kalai-Linial theorem, Friedgut’s junta theorem and the invariance principle of Mossel, O’Donnell and Oleszkiewicz. In these results the cube is equipped with the uniform ( 1 / 2 1/2 -biased) measure, but it is desirable, particularly for applications to the theory of sharp thresholds, to also obtain such results for general p p -biased measures. However, simple examples show that when p p is small there is no hypercontractive inequality that is strong enough for such applications. In this paper, we establish an effective hypercontractivity inequality for general p p that applies to ‘global functions’, i.e. functions that are not significantly affected by a restriction of a small set of coordinates. This class of functions appears naturally, e.g. in Bourgain’s sharp threshold theorem, which states that such functions exhibit a sharp threshold. We demonstrate the power of our tool by strengthening Bourgain’s theorem, making progress on two conjectures of Kahn and Kalai (both these conjectures were open when we arXived this paper in 2019; one of them was solved in 2022; the other is still open), and proving a p p -biased analogue of the seminal invariance principle of Mossel, O’Donnell, and Oleszkiewicz. In this 2023 version of our paper we will also survey many further applications of our results that have been obtained by various authors since we arXived the first version in 2019.

Nilspace Factors for General Uniformity Seminorms, Cubic Exchangeability and Limits

We study a class of measure-theoretic objects that we call cubic couplings, on which there is a common generalization of the Gowers norms and the Host–Kra seminorms. Our main result yields a complete structural description of cubic couplings, using nilspaces. We give three applications. Firstly, we describe the characteristic factors of Host–Kra type seminorms for measure-preserving actions of countable nilpotent groups. This yields an extension of the structure theorem of Host and Kra. Secondly, we characterize sequences of random variables with a property that we call cubic exchangeability. These are sequences indexed by the infinite discrete cube, such that for every integer k ≥ 0 k\geq 0 the joint distribution’s marginals on affine subcubes of dimension k k are all equal. In particular, our result gives a description, in terms of compact nilspaces, of a related exchangeability property considered by Austin, inspired by a problem of Aldous. Finally, using nilspaces we obtain limit objects for sequences of functions on compact abelian groups (more generally on compact nilspaces) such that the densities of certain patterns in these functions converge. The paper thus proposes a measure-theoretic framework on which the area of higher-order Fourier analysis can be based, and which yields new applications of this area in a unified way in ergodic theory and arithmetic combinatorics.

On the $$\Phi $$-stability and related conjectures

Given a convex function $$\Phi :[0,1]\rightarrow {\mathbb {R}}$$ and the mean $${\mathbb {E}}f({\textbf{X}})=a\in [0,1]$$ , which Boolean function f maximizes the $$\Phi $$ -stability $${\mathbb {E}}[\Phi (T_{\rho }f({\textbf{X}}))]$$ of f? Here $${\textbf{X}}$$ is a random vector uniformly distributed on the discrete cube $$\{-1,1\}^{n}$$ and $$T_{\rho }$$ is the Bonami–Beckner operator. Special cases of this problem include the (symmetric and asymmetric) $$\alpha $$ -stability problems and the “Most Informative Boolean Function” problem. In this paper, we provide several upper bounds for the maximal $$\Phi $$ -stability. When specializing $$\Phi $$ to some particular forms, by these upper bounds, we partially resolve Mossel and O’Donnell’s conjecture on $$\alpha $$ -stability with $$\alpha >2$$ , Li and Médard’s conjecture on $$\alpha $$ -stability with $$1<\alpha <2$$ , and Courtade and Kumar’s conjecture on the “Most Informative Boolean Function” which corresponds to a conjecture on $$\alpha $$ -stability with $$\alpha =1$$ . Our proofs are based on discrete Fourier analysis, optimization theory, and improvements of the Friedgut–Kalai–Naor (FKN) theorem. Our improvements of the FKN theorem are sharp or asymptotically sharp for certain cases.

Equal sums in random sets and the concentration of divisors

We study the extent to which divisors of a typical integer n are concentrated. In particular, defining Delta (n) := max _t # {d | n, log d in [t,t+1]}, we show that Delta (n) geqslant (log log n)^{0.35332277ldots } for almost all n, a bound we believe to be sharp. This disproves a conjecture of Maier and Tenenbaum. We also prove analogs for the concentration of divisors of a random permutation and of a random polynomial over a finite field. Most of the paper is devoted to a study of the following much more combinatorial problem of independent interest. Pick a random set {textbf{A}} subset {mathbb {N}} by selecting i to lie in {textbf{A}} with probability 1/i. What is the supremum of all exponents beta _k such that, almost surely as D rightarrow infty , some integer is the sum of elements of {textbf{A}} cap [D^{beta _k}, D] in k different ways? We characterise beta _k as the solution to a certain optimisation problem over measures on the discrete cube {0,1}^k, and obtain lower bounds for beta _k which we believe to be asymptotically sharp.

Entropic curvature on graphs along Schrödinger bridges at zero temperature

Lott-Sturm-Villani theory of curvature on geodesic spaces has been extended to discrete graph spaces by C. L{\'e}onard by replacing W2-Wasserstein geodesics by Schr{\"o}odinger bridges in the definition of entropic curvature [23, 25, 24]. As a remarkable fact, as a temperature parameter goes to zero, these Schr{\"o}dinger bridges are supported by geodesics of the space. We analyse this property on discrete graphs to reach entropic curvature on discrete spaces. Our approach provides lower bounds for the entropic curvature for several examples of graph spaces: the lattice Z n endowed with the counting measure, the discrete cube endowed with product probability measures, the circle, the complete graph, the Bernoulli-Laplace model. Our general results also apply to a large class of graphs which are not specifically studied in this paper. As opposed to Erbar-Maas results on graphs [27, 10, 11], entropic curvature results of this paper imply new Pr{\'e}kopa-Leindler type of inequalities on discrete spaces, and new transport-entropy inequalities related to refined concentration properties for the graphs mentioned above. For example on the discrete hypercube {0, 1} n and for the Bernoulli Laplace model, a new W2 -- W1 transport-entropy inequality is reached, that can not be derived by usual induction arguments over the dimension n. As a surprising fact, our method also gives improvements of weak transport-entropy inequalities (see [28, 15]) associated to the so-called convex-hull method by Talagrand [38].

Sharp threshold for the Ising perceptron model

Consider the discrete cube {−1,1}N and a random collection of half spaces which includes each half space H(x):={y∈{−1,1}N:x·y≥κ N} for x∈{−1,1}N independently with probability p. Is the intersection of these half spaces empty? This is called the Ising perceptron model under Bernoulli disorder. We prove that this event has a sharp threshold, that is, the probability that the intersection is empty increases quickly from ϵ to 1−ϵ when p increases only by a factor of 1+o(1) as N→∞.

A new look at random projections of the cube and general product measures

A consequence of the celebrated Dvoretzky–Milman theorem is a strong law of large numbers for d-dimensional random projections of the n-dimensional cube. It shows that, with respect to the Hausdorff distance, a uniform random projection of the cube [−1/n,+1/n]n onto Rd converges almost surely to a centered d-dimensional Euclidean ball of radius 2/π, as n→∞. We start by providing an alternative proof of this strong law via the Artstein–Vitale law of large numbers for random compact sets. Then, for every point inside the ball of radius 2/π, we determine the asymptotic number of vertices and the volume of the part of the cube projected ‘close’ to this point. More generally, we study large deviations for random projections of arbitrary product measures. Let ν⊗n be the n-fold product measure of a Borel probability measure ν on R, and let I be uniformly distributed on the Stiefel manifold of orthogonal d-frames in Rn. It is shown that the sequence of random measures ν⊗n∘(n−1/2I∗)−1, n∈N, satisfies a large deviation principle with probability 1. The rate function is explicitly identified in terms of the moment generating function of ν. At the heart of the proofs lies a transition trick which allows to replace the uniform projection by the Gaussian one. A number of concrete examples are discussed as well, including the uniform distributions on the cube [−1,1]n and the discrete cube {−1,1}n as special cases.

Anisotropic bootstrap percolation in three dimensions

Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^{3}$, and assume that the neighborhood of each vertex consists of the $a_{i}$ nearest neighbors in the $\pm e_{i}$-directions for each $i\in \{1,2,3\}$, where $a_{1}\le a_{2}\le a_{3}$. Suppose we infect any healthy vertex $x\in [L]^{3}$ already having $a_{3}+1$ infected neighbors, and that infected sites remain infected forever. In this paper, we determine the critical length for percolation up to a constant factor in the exponent, for all triples $(a_{1},a_{2},a_{3})$. To do so, we introduce a new algorithm called the beams process and prove an exponential decay property for a family of subcritical two-dimensional bootstrap processes.

Rademacher type and Enflo type coincide

A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure. We prove that Rademacher type and Enflo type coincide, settling a long-standing open problem in Banach space theory. The proof is based on a novel dimension-free analogue of Pisier's inequality on the discrete cube.

A Note on Norine's Antipodal-Colouring Conjecture

Norine's antipodal-colouring conjecture, in a form given by Feder and Subi, asserts that whenever the edges of the discrete cube are 2-coloured there must exist a path between two opposite vertices along which there is at most one colour change. The best bound to date was that there must exist such a path with at most $n/2$ colour changes. Our aim in this note is to improve this upper bound to $(\frac{3}{8}+o(1))n$.

No More Than $$2^{d+1}-2$$ Nearly Neighbourly Simplices in $${\mathbb {R}}^d$$

We prove a combinatorial theorem on families of disjoint sub-boxes of a discrete cube, which implies that there are at most $$2^{d+1}-2$$ nearly neighbourly simplices in $$\mathbb {R}^d$$ .

Second order concentration via logarithmic Sobolev inequalities

We show sharpened forms of the concentration of measure phenomenon centered at first order stochastic expansions. The bound are based on second order difference operators and second order derivatives. Applications to functions on the discrete cube and stochastic Hoeffding type expansions in mathematical statistics are studied as well as linear eigenvalue statistics in random matrix theory.

Talagrand Inequality at Second Order and Application to Boolean Analysis

This note is concerned with an extension, at second order, of an inequality on the discrete cube $$C_n=\{-\,1,1\}^n$$ (equipped with the uniform measure) due to Talagrand (Ann Probab 22:1576–1587, 1994). As an application, the main result of this note is a theorem in the spirit of a famous result from Kahn et al. (cf. Proceedings of 29th Annual Symposium on Foundations of Computer Science, vol 62. Computer Society Press, Washington, pp 68–80, 1988) concerning the influence of Boolean functions. The notion of the influence of a couple of coordinates $$(i,j)\in \{1,\ldots ,n\}^2$$ is introduced in Sect. 2, and the following alternative is obtained: For any Boolean function $$f\,:\, C_n\rightarrow \{0,1\}$$, either there exists a coordinate with influence at least of order $$(1/n)^{1/(1+\eta )}$$, with $$\, 0<\eta <1$$ (independent of f and n), or there exists a couple of coordinates $$(i,j)\in \{1,\ldots ,n\}^2$$ with $$i\ne j$$, with influence at least of order $$(\log n/n)^2$$. In Sect. 4, it is shown that this extension of Talagrand inequality can also be obtained, with minor modifications, for the standard Gaussian measure $$\gamma _n$$ on $${\mathbb {R}}^n$$; the obtained inequality can be of independent interest. The arguments rely on interpolation methods by semigroup together with hypercontractive estimates. At the end of the article, some related open questions are presented.

Semi-perfect 1-factorizations of the hypercube

A 1-factorization M={M1,M2,…,Mn} of a graph G is called perfect if the union of any pair of 1-factors Mi,Mj with i≠j is a Hamilton cycle. It is called k-semi-perfect if the union of any pair of 1-factors Mi,Mj with 1≤i≤k and k+1≤j≤n is a Hamilton cycle.We consider 1-factorizations of the discrete cube Qd. There is no perfect 1-factorization of Qd, but it was previously shown that there is a 1-semi-perfect 1-factorization of Qd for all d. Our main result is to prove that there is a k-semi-perfect 1-factorization of Qd for all k and all d, except for one possible exception when k=3 and d=6. This is, in some sense, best possible.We conclude with some questions concerning other generalizations of perfect 1-factorizations.

On a biased edge isoperimetric inequality for the discrete cube

The ‘full’ edge isoperimetric inequality for the discrete cube {0,1}n (due to Harper, Lindsey, Berstein and Hart) specifies the minimum size of the edge boundary ∂A of a set A⊂{0,1}n, as function of |A|. A weaker (but more widely-used) lower bound is |∂A|≥|A|log⁡(2n/|A|), where equality holds whenever A is a subcube. In 2011, the first author obtained a sharp ‘stability’ version of the latter result, proving that if |∂A|≤|A|(log⁡(2n/|A|)+ϵ), then there exists a subcube C such that |AΔC|/|A|=O(ϵ/log⁡(1/ϵ)).The ‘weak’ version of the edge isoperimetric inequality has the following well-known generalization for the ‘p-biased’ measure μp on the discrete cube: if p≤1/2, or if 0<p<1 and A is monotone increasing, then pμp(∂A)≥μp(A)logp⁡(μp(A)).In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if pμp(∂A)≤μp(A)(logp⁡(μp(A))+ϵ), then there exists a subcube C such that μp(AΔC)/μp(A)=O(ϵ′/log⁡(1/ϵ′)), where ϵ′:=ϵln⁡(1/p). This result is a central component in recent work of the authors proving sharp stability versions of a number of Erdős–Ko–Rado type theorems in extremal combinatorics, including the seminal ‘complete intersection theorem’ of Ahlswede and Khachatrian.In addition, we prove a biased-measure analogue of the ‘full’ edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the ‘full’ edge isoperimetric inequality, one relying on the Kruskal–Katona theorem.

Gaussian-width gradient complexity, reverse log-Sobolev inequalities and nonlinear large deviations

We prove structure theorems for measures on the discrete cube and on Gaussian space, which provide sufficient conditions for mean-field behavior. These conditions rely on a new notion of complexity for such measures, namely the Gaussian-width of the gradient of the log-density. On the cube {−1, 1}n, we show that a measure $${\nu}$$ which exhibits low complexity can be written as a mixtureof measures $${\{\nu_\theta\}_{\theta\in\mathcal{I}}}$$ such that: (i) for each $${\theta}$$ , the measure $${\nu_\theta}$$ is a small perturbation of $${\nu}$$ such that log $$\frac{{d \nu_\theta}}{{d \nu}}$$ is a linear function whose gradient is small and, (ii) $${\nu_\theta}$$ is close to some product measure, in Wasserstein distance, for most $${\theta}$$ . Thus, our framework can be used to study the behavior of low-complexity measures beyond approximation of the partition function, showing that those measures are roughly mixtures of product measures whose entropy is close to that of the original measure. In particular, as a corollary of our theorems, we derive a bound for the naive mean-field approximation of the log-partition function which improves the nonlinear large deviation framework of Chatterjee and Dembo (Adv Math, 319:313–347, 2017. ISSN 0001-8708. https://dx.doi.org/10.1016/j.aim.2017.08.003 ) in several ways: (1) It does not require any bounds on second derivatives. (2) The covering number is replaced by the weaker notion of Gaussian-width. (3) We obtain stronger asymptotics with respect to the dimension. Two other corollaries are decomposition theorems for exponential random graphs and large-degree Ising models. In the Gaussian case, we show that measures of low-complexity exhibit an almost-tight reverse log-Sobolev inequality.

On the structure of subsets of the discrete cube with small edge boundary

On the structure of subsets of the discrete cube with small edge boundary, Discrete Analysis 2018:9, 29 pp. An isoperimetric inequality is a statement that tells us how small the boundary of a set can be given the of the set, for suitable notions of size and boundary. For example, one formulation of the classical isoperimetric inequality in $\mathbb R^n$ is as follows. Given a subset $X$ of $\mathbb R^n$, define the $\epsilon$-_expansion_ of $X$ to be the open set $X_\epsilon=\{y\in\mathbb R^n: d(y,X)<\epsilon\}$. If in addition $X$ is measurable, define the of its boundary to be $\lim\inf_{\epsilon\to 0} \epsilon^{-1}\mu(X_\epsilon\setminus X)$. (If $X$ is a set with a suitably smooth topological boundary $\partial X$, then this turns out to equal the surface measure of $\partial X$.) Then amongst all sets $X$ of a given measure, the one with the smallest boundary is an $n$-dimensional ball. Isoperimetric inequalities have been the focus of a great deal of research, partly for their intrinisic interest, but also because they have numerous applications. One particularly useful one is the _edge-isoperimetric inequality in the discrete cube_. This concerns subsets $X$ of the $n$-dimensional cube $\{0,1\}^n$, which we turn into a graph by joining two points $x$ and $y$ if they differ in exactly one coordinate. The of a set $X$ is simply its cardinality, the _edge-boundary_ of $X$ is defined to be the set of edges between $X$ and its complement, and the of the edge-boundary is the number of such edges. If $|X|=2^d$, then it is known that the edge-boundary is minimized when $X$ is a $d$-dimensional subspace of $\mathbb F_2^n$ generated by $d$ standard basis vectors. More generally, if $|X|=m$, then the edge-boundary is minimized when $X$ is an initial segment in the lexicographical ordering, which is the ordering where we set $x<y$ if $x_i<y_i$ for the first coordinate $i$ where $x_i$ and $y_i$ differ. (This coincides with the ordering we obtain if we think of the sequences as binary representations of integers.) For the two isoperimetric inequalities just mentioned, as well as many others, it is known that the extremal examples provided are essentially the only ones: a subset of $\mathbb R^n$ with a boundary that is as small as possible has to be an $n$-dimensional ball, and a subset of the discrete cube with edge-boundary that is as small as possible has to be an initial segment of the lexicographical ordering, up to the symmetries of the graph. Furthermore, there are _stablity_ results: a set with a boundary that is _almost_ as small as possible must be close to an extremal example. Such a result tells us that the isoperimetric inequalities are robust, in the sense that if you slightly perturb the condition on the set, then you only slightly perturb what the set has to look like. This paper is about an extremely precise stability result for the edge isoperimetric inequality in the discrete cube. There have been a number of papers on such results (see the introduction to the paper for details), but they have been mainly for sets of $2^d$ for some $d$, where the goal is to prove that they must be close to $d$-dimensional subcubes -- that is, subspaces (or their translates) generated by $d$ standard basis vectors. This paper considers sets of arbitrary and proves the following result. Suppose that $X$ is a subset of $\{0,1\}^n$ of $m$. Suppose that the of the edge-boundary of $X$ is at most $g_n(m)+l$, where $g_n(m)$ is the of the edge-boundary of the initial segment $I_m$ of $m$ in the lexicographical order. Then there is an automorphism $\phi$ of $\{0,1\}^n$ (meaning a bijection that takes neighbouring points to neighbouring points) such that $|X\Delta\phi(I_m)|\leq Cl$, where $C$ is an absolute constant. They give an example to show that $C$ must be at least 2, and thus that their result is best possible up to the value of the constant $C$. Previous proofs of stability versions of the edge-isoperimetric inequality in the cube have used Fourier analysis. The proof in this paper uses purely combinatorial methods, such as induction on the dimension, and compressions. To get these methods to work, several interesting ideas are needed, including some new results about the influence of variables.

An isoperimetric inequality for antipodal subsets of the discrete cube

We say a family of subsets of {1,2,…,n} is antipodal if it is closed under taking complements. We prove a best-possible isoperimetric inequality for antipodal families of subsets of {1,2,…,n} (of any size). Our inequality implies that for any k∈N, among all such families of size 2k, a family consisting of the union of two antipodal (k−1)-dimensional subcubes has the smallest possible edge boundary.

Http://discreteanalysisjournal.com/article/2105-notes-on-nilspaces-algebraic-aspects

Notes on nilspaces: algebraic aspects, Discrete Analysis 2017:15, 59 pp. One of the fundamental insights in modern additive combinatorics is that there is a hierarchy of notions of "pseudorandomness" or "higher order Fourier uniformity" that can be applied either to subsets $A$ of an abelian group $G$, or functions $f: G \to {\bf C}$ of that abelian group. For instance, to understand the pseudorandomness of a subset $A$ of an abelian group $G$, one can count the number of parallelograms $$ (x, x+h_1, x+h_2, x+h_1+h_2)$$ that are fully contained in $A$, or (for a higher notion of pseudorandomness) instead count parallelepipeds $ (x, x+h_1, x+h_2, x+h_1+h_2, x+h_3, x+h_1+h_3,$ $x+h_2+h_3, x+h_1+h_2+h_3)$ or even higher-dimensional parallelepipeds. It turns out that the study of these parallelepipeds is of interest in its own right. For each dimension $d$, let $G^{[d]}$ denote the space of $d$-dimensional parallelepipeds in $G$; this is a certain subgroup of $G^{2^d}$. These spaces interact with each other in a number of ways: for instance, each $d-1$-dimensional face of the discrete cube $\{0,1\}^d$ induces a restriction map from $G^{[d]}$ to $G^{[d-1]}$. For instance, if $(x_1,\dots,x_8)$ lies in $G^{[3]}$, then $(x_1,x_2,x_3,x_4)$ or $(x_1,x_2,x_5,x_6)$ will lie in $G^{[2]}$. In the converse direction, we have the _corner completion_ property: if for instance one has seven vertices $(x_1,\dots,x_7)$ in $G$, with the property that all the two-dimensional subfaces such as $(x_1,x_2,x_3,x_4)$ or $(x_1,x_2,x_5,x_6)$ lie in $G^{[2]}$, then one can find an additional element $x_8$ of $G$ such that $(x_1,\dots,x_8)$ lies in $G^{[3]}$ (indeed, in this case this additional element will be unique). In papers of Host-Kra (["Parallelepipeds, Nilpotent Groups, and Gowers Norms"](https://arxiv.org/abs/math/0606004)) and Camarena-Szegedy (["Nilspaces, nilmanifolds and their morphisms"](https://arxiv.org/abs/1009.3825)), these properties of parallelepipeds were abstracted into axioms for a new type of structure, referred to as parallelepiped structures in Host-Kra and nilspaces in Camarena-Szegedy. In addition to the example given above of parallelepipeds on an abelian group $G$, another fundamental example of these structures comes from _nilmanifolds_ $G/\Gamma$, formed by quotienting a nilpotent Lie group $G$ by a lattice $\Gamma$. For instance, the analogue of parallelograms in this setting would be quadruples of the form $$ (x, g_1 x, g'_1 x, g_1 g'_1 g_2 x)$$ for $x \in G/\Gamma$, $g_1,g'_1 \in G$, and $g_2$ in the commutator subgroup $[G,G]$. Such spaces emerged naturally in an ergodic-theory context in work of Host and Kra (["Nonconventional ergodic averages and nilmanifolds"](http://annals.math.princeton.edu/wp-content/uploads/annals-v161-n1-p08.pdf)) and in a (nonstandard analysis) combinatorial context in work of Szegedy (["On higher order Fourier analysis"](https://arxiv.org/abs/1203.2260)), so it became natural to develop a systematic theory of such spaces, and in particular to look for a satisfactory classification of these spaces. As it turns out, the theory can be divided into two parts. The first part, which is simpler, is the purely "algebraic" theory of nilspaces, in which one does not require that the parallepiped structures are compatible in any way with a measure-theoretic or topological structure on the space. Then there is the "topological" and "measure-theoretic" part of the theory, in which one studies how the parallelepipeds interact with these structures. This division is analogous to the distinction between abstract group theory, and the theory of topological groups (and their interaction with measure-theoretic concepts such as Haar measure). In this paper, the first in a two-part series, the author systematically lays out the algebraic theory of nilspaces. The discussion follows closely the earlier work of Camarena and Szegedy, but with more detailed proofs and additional discussion of key examples. Just as nilpotent groups (or nilmanifolds) can be arranged in a hierarchy depending on the "step" or "nilpotency class" of the underlying nilpotent group, one can also assign a notion of a "step" to a nilspace, with higher step nilspaces being more complicated than lower step ones. For instance, as is shown in this paper, 1-step nilspaces are essentially the same as abelian groups. Perhaps the deepest result established in this paper is the fact that a $k$-step nilspace can always be viewed as an "abelian extension" of a $k-1$-step nilspace, in much the same way that a $k$-step nilpotent group can be viewed as a central extension of a $k-1$-step nilpotent group, furthermore, one can associate with the extension a certain "cocycle" which determines the $k$-step nilspace completely (up to isomorphism) once the underlying $k-1$-step nilspace is specified. Conversely, every such cocycle will generate such a $k$-step nilspace. This machinery will be used in [the second part of the series](http://discreteanalysisjournal.com/article/2106-notes-on-compact-nilspaces) to study compact nilspaces.

Noise stability and correlation with half spaces

Benjamini, Kalai and Schramm showed that a monotone function $f : \{-1,1\}^n \to \{-1,1\}$ is noise stable if and only if it is correlated with a half-space (a set of the form $\{x: \langle x, a \rangle \le b\}$). We study noise stability in terms of correlation with half-spaces for general (not necessarily monotone) functions. We show that a function $f: \{-1, 1\}^n \to \{-1, 1\}$ is noise stable if and only if it becomes correlated with a half-space when we modify $f$ by randomly restricting a constant fraction of its coordinates. Looking at random restrictions is necessary: we construct noise stable functions whose correlation with any half-space is $o(1)$. The examples further satisfy that different restrictions are correlated with different half-spaces: for any fixed half-space, the probability that a random restriction is correlated with it goes to zero. We also provide quantitative versions of the above statements, and versions that apply for the Gaussian measure on $\mathbb{R} ^n$ instead of the discrete cube. Our work is motivated by questions in learning theory and a recent question of Khot and Moshkovitz.