Higher-order Fourier Analysis Research Articles

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 higher-order Fourier analysis in characteristic p

AbstractIn this paper, the nilspace approach to higher-order Fourier analysis is developed in the setting of vector spaces over a prime field $\mathbb {F}_p$ , with applications mainly in ergodic theory. A key requisite for this development is to identify a class of nilspaces adequate for this setting. We introduce such a class, whose members we call p-homogeneous nilspaces. One of our main results characterizes these objects in terms of a simple algebraic property. We then prove various further results on these nilspaces, leading to a structure theorem describing every finite p-homogeneous nilspace as the image, under a nilspace fibration, of a member of a simple family of filtered finite abelian p-groups. The applications include a description of the Host–Kra factors of ergodic $\mathbb {F}_p^\omega $ -systems as p-homogeneous nilspace systems. This enables the analysis of these factors to be reduced to the study of such nilspace systems, with central questions on the factors thus becoming purely algebraic problems on finite nilspaces. We illustrate this approach by proving that for $k\leq p+1$ the kth Host–Kra factor is an Abramov system of order at most k, extending a result of Bergelson–Tao–Ziegler that holds for $k< p$ . We illustrate the utility of p-homogeneous nilspaces also by showing that the structure theorem yields a new proof of the Tao–Ziegler inverse theorem for Gowers norms on $\mathbb {F}_p^n$ .

Stabilizer rank and higher-order Fourier analysis

We establish a link between stabilizer states, stabilizer rank, and higher-order Fourier analysis – a still-developing area of mathematics that grew out of Gowers's celebrated Fourier-analytic proof of Szemerédi's theorem \cite{gowers1998new}. We observe that n-qudit stabilizer states are so-called nonclassical quadratic phase functions (defined on affine subspaces of Fpn where p is the dimension of the qudit) which are fundamental objects in higher-order Fourier analysis. This allows us to import tools from this theory to analyze the stabilizer rank of quantum states. Quite recently, in \cite{peleg2021lower} it was shown that the n-qubit magic state has stabilizer rank Ω(n). Here we show that the qudit analog of the n-qubit magic state has stabilizer rank Ω(n), generalizing their result to qudits of any prime dimension. Our proof techniques use explicitly tools from higher-order Fourier analysis. We believe this example motivates the further exploration of applications of higher-order Fourier analysis in quantum information theory.

The List Decoding Radius for Reed–Muller Codes Over Small Fields

The list decoding problem for a code asks for the maximal radius up to which any ball of that radius contains only a constant number of codewords. The list decoding radius is not well understood even for well studied codes like Reed–Solomon or Reed–Muller codes. Fix a finite field ${\mathbb {F}}$ . The Reed–Muller code $\text {RM}_{ {\mathbb {F}}}(n,d)$ is defined by $n$ -variate degree- $d$ polynomials over ${\mathbb {F}}$ . In this paper, we study the list decoding radius of Reed–Muller codes over a constant prime field ${\mathbb {F}}= {\mathbb {F}}_{p}$ , constant degree $d$ , and large $n$ . We show that the list decoding radius is equal to the minimal distance of the code. That is, if we denote by ${\delta }(d)$ the normalized minimal distance of $\text {RM}_{ {\mathbb {F}}}(n,d)$ , then the number of codewords in any ball of radius ${\delta }(d)- {\varepsilon }$ is bounded by $c=c(p,d, {\varepsilon })$ independent of $n$ . This resolves a conjecture of Gopalan et al. , who among other results proved it in the special case of ${\mathbb {F}}= {\mathbb {F}}_{2}$ ; and extends the work of Gopalan who proved the conjecture in the case of $d=2$ . We also analyse the number of codewords in balls of radius exceeding the minimal distance of the code. For $e \leq d$ , we show that the number of codewords of $\text {RM}_{ {\mathbb {F}}}(n,d)$ in a ball of radius ${\delta }(e) - {\varepsilon }$ is bounded by $\exp (c \cdot n^{d-e})$ , where $c=c(p,d, {\varepsilon })$ is independent of $n$ . The dependence on $n$ is tight. This extends the work of Kaufman et al. who proved similar bounds over ${\mathbb {F}}_{2}$ . The proof relies on several new ingredients: an extension of the Frieze–Kannan weak regularity to general function spaces, higher order Fourier analysis, and an extension of the Schwartz–Zippel lemma to the compositions of polynomials.

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.

Notes on compact nilspaces

Notes on compact Discrete Analysis 2017:16, 57 pp. This is the second paper in a two-part series. The first paper, [also published in this journal](http://discreteanalysisjournal.com/article/2105-notes-on-nilspaces-algebraic-aspects), developed the algebraic theory of nilspaces, which are an abstraction both of the notion of a nilmanifold, and of that of an abelian group (both with their attendant parallelogram structures). In this paper, the author applies this algebraic theory to study compact which can be viewed as a generalization of the notion of a compact abelian group. The main motivation for this is that such compact nilspaces arise naturally when trying to attack additive combinatorial problems such as the inverse conjecture for the Gowers norms via the methods of nonstandard analysis (or ultraproducts), as was demonstrated in work of Szegedy ([On higher order Fourier analysis](https://arxiv.org/abs/1203.2260)). The algebraic theory in the first part lets one describe $k$-step compact nilspaces as abelian extensions of $k-1$-step with the extension given by a cocycle. Among other things, this can be used to assign to each nilspace a probability measure that is analogous to the Haar probability measure one can assign to a compact group. However, it is not immediately obvious that this cocycle has any good measurability or continuity properties. Much of the delicate analysis in this paper (particularly those concerning the properties of continuous systems of measures) is devoted to ensuring that such good properties can be imposed (locally at least) on the cocycle, perhaps after modifying the cocycle by a suitable coboundary that does not affect the isomorphism class of the extension. Indeed, one can view the paper as developing some basic for although the formidable machinery of modern cohomology is not really deployed here. As with the first paper in this series, the paper closely follows the previous work of Camarena and Szegedy, but with many more details provided in the arguments. (An alternative approach to this theory has also recently been worked out in a series of papers by Gutman, Manners, and Varju.) A full structural description of compact nilspaces seems to be quite difficult, but in this paper two important partial descriptions are obtained. The first is to describe every compact nilspace as an inverse limit of compact nilspaces of rank. This is somewhat analogous to how the Peter-Weyl theorem can be used to describe a compact group as the inverse limit of finite-dimensional compact Lie groups. The second is to show that every connected compact nilspace of finite rank is isomorphic to a nilmanifold. These results were previously obtained by Camarena and Szegedy (and later used by Szegedy to provide a new proof of the inverse conjecture for the Gowers uniformity norms), but the author here reproves these results with much more detailed proofs, in particular treating with some care certain steps that were only briefly discussed in the prior paper of Camarena and Szegedy.

Lower Bounds for Constant Query Affine-Invariant LCCs and LTCs

Affine-invariant codes are codes whose coordinates form a vector space over a finite field and which are invariant under affine transformations of the coordinate space. They form a natural, well-studied class of codes; they include popular codes such as Reed-Muller and Reed-Solomon. A particularly appealing feature of affine-invariant codes is that they seem well suited to admit local correctors and testers. In this work, we give lower bounds on the length of locally correctable and locally testable affine-invariant codes with constant query complexity. We show that if a code C ⊂ Σ K n is an r -query affine invariant locally correctable code (LCC), where K is a finite field and Σ is a finite alphabet, then the number of codewords in C is at most exp( O K,r,|Σ| ( n r −1)) . Also, we show that if C ⊂ Σ K n is an r -query affine invariant locally testable code (LTC), then the number of codewords in C is at most exp( O K,r,|Σ| ( n r −2)) . The dependence on n in these bounds is tight for constant-query LCCs/LTCs, since Guo, Kopparty, and Sudan (ITCS’13) constructed affine-invariant codes via lifting that have the same asymptotic tradeoffs. Note that our result holds for non-linear codes, whereas previously, Ben-Sasson and Sudan (RANDOM’11) assumed linearity to derive similar results. Our analysis uses higher-order Fourier analysis. In particular, we show that the codewords corresponding to an affine-invariant LCC/LTC must be far from each other with respect to Gowers norm of an appropriate order. This then allows us to bound the number of codewords, using known decomposition theorems, which approximate any bounded function in terms of a finite number of low-degree non-classical polynomials, up to a small error in the Gowers norm.

General systems of linear forms: Equidistribution and true complexity

Higher-order Fourier analysis is a powerful tool that can be used to analyze the densities of linear systems (such as arithmetic progressions) in subsets of Abelian groups. We are interested in the group Fpn, for fixed p and large n, where it is known that analyzing these averages reduces to understanding the joint distribution of a family of sufficiently pseudorandom (formally, high-rank) nonclassical polynomials applied to the corresponding system of linear forms.In this work, we give a complete characterization for these distributions for arbitrary systems of linear forms. This extends previous works which accomplished this in some special cases. As an application, we resolve a conjecture of Gowers and Wolf on the true complexity of linear systems. Our proof deviates from that of the previously known special cases and requires several new ingredients. One of which, which may be of independent interest, is a new theory of homogeneous nonclassical polynomials.

Structure theorems in additive combinatorics

Several structure results of additive combinatorics are considered. Classical and modern problems of combinatorial number theory, higher-order Fourier analysis, inverse theorems for Gowers norms, higher energies, and the relationship between combinatorial and analytic number theory are discussed. Bibliography: 149 titles.

Limits of Boolean Functions on $\mathbb{F}_p^n$

We study sequences of functions of the form $\mathbb{F}_p^n \to \{0,1\}$ for varying $n$, and define a notion of convergence based on the induced distributions from restricting the functions to a random affine subspace. Using a decomposition theorem and a recently proven equi-distribution theorem from higher order Fourier analysis, we prove that the limits of such convergent sequences can be represented by certain measurable functions. We also show that every such limit object arises as the limit of some sequence of functions. These results are in the spirit of similar results which have been developed for limits of graph sequences. A more general, albeit substantially more sophisticated, limit object was recently constructed by Balázs Szegedy [Gowers norms, regularization and limits of functions on abelian groups. 2010. arXiv:1010.6211].

Convergence Results for Systems of Linear Forms on Cyclic Groups and Periodic Nilsequences

Given a positive integer $N$ and real number $\alpha\in [0, 1]$, let $m(\alpha,N)$ denote the minimum, over all sets $A\subseteq \mathbb{Z}_{N}$ of size at least $\alpha N$, of the normalized count of 3-term arithmetic progressions contained in $A$. A theorem of Croot states that $m(\alpha,N)$ converges as $N\to\infty$ through the primes, answering a question of Green. Using recent advances in higher-order Fourier analysis, we prove an extension of this theorem, showing that the result holds for $k$-term progressions for general $k$ and further for all systems of integer linear forms of finite complexity. We also obtain a similar convergence result for the maximum densities of sets free of solutions to systems of linear equations. These results rely on a regularity method for functions on finite cyclic groups that we frame in terms of periodic nilsequences, using in particular some regularity results of Szegedy (relying on his joint work with Camarena) and the equidistribution results of Green and Tao.

Correlation Testing for Affine Invariant Properties on $\mathbb{F}_p^n$ in the High Error Regime

Recently there has been much interest in Gowers uniformity norms from the perspective of theoretical computer science. This is mainly due to the fact that these norms provide a method for testing whether the maximum correlation of a function $f:\mathbb{F}_p^n\to\mathbb{F}_p$ with polynomials of degree at most $d\leq p$ is nonnegligible, while making only a constant number of queries to the function. This is an instance of correlation testing. In this framework, a fixed test is applied to a function, and the acceptance probability of the test is dependent on the correlation of the function from the property. This is an analogue of proximity oblivious testing, a notion coined by Goldreich and Ron, in the high error regime. In this work, we study general properties which are affine invariant and which are correlation testable using a constant number of queries. We show that any such property (as long as the field size is not too small) can in fact be tested by Gowers uniformity tests, and hence having correlation with the property is equivalent to having correlation with degree $d$ polynomials for some fixed $d$. We stress that our result holds also for nonlinear properties which are affine invariant. This completely classifies affine invariant properties which are correlation testable. The proof is based on higher-order Fourier analysis. Another ingredient is a nontrivial extension of a graph theoretical theorem of Erdös, Lovász, and Spencer to the context of additive number theory.

Terence Tao: “Higher Order Fourier Analysis”

Terence Tao: “Higher Order Fourier Analysis”