arXiv subjects

Diego González-Sánchez

Publications and source records attributed to Diego González-Sánchez.

18 recordsLinked to original sources

arXiv2026

An inverse theorem for all finite abelian groups via nilmanifolds

We prove a first inverse theorem for Gowers norms on all finite abelian groups that uses only nilmanifolds (rather than possibly more general nilspaces). This makes progress toward confirming the Jamneshan--Tao conjecture. The correlating function in our theorem is a projected nilsequence, obtained as the fiber-wise average of a nilsequence defined on a boundedly-larger abelian group extending the original abelian group. This result is tight in the following sense: we prove also that $k$-step projected nilsequences of bounded complexity are genuine obstructions to having small Gowers $U^{k+1}$-norm. This inverse theorem relies on a new result concerning compact finite-rank (CFR) nilspaces, which is the main contribution in this paper: every $k$-step CFR nilspace is a factor of a $k$-step nilmanifold. This new connection between the classical theory of nilmanifolds and the more recent theory of nilspaces has applications beyond arithmetic combinatorics. We illustrate this with an application in topological dynamics, by proving the following result making progress on a question of Jamneshan, Shalom and Tao: every minimal $\mathbb{Z}^ω$-system of order $k$ is a factor of an inverse limit of $\mathbb{Z}^ω$-polynomial orbit systems of order $k$, these being natural generalizations of nilsystems alternative to translational systems.

math.DS↗

arXiv2026

The Jamneshan-Tao conjecture for finite abelian groups of bounded rank

We confirm the Jamneshan-Tao conjecture for finite abelian groups of rank at most a fixed integer $R$ (i.e. finite abelian groups generated by at most $R$ elements), by proving an inverse theorem for 1-bounded functions of non-trivial Gowers norm on such groups, concluding that such a function must correlate non-trivially with a nilsequence of bounded complexity.

math.GR↗

arXiv2025

Free nilspaces, double-coset nilspaces, and Gowers norms

Compact finite-rank nilspaces have become central in the nilspace approach to higher-order Fourier analysis, notably through their role in a general form of the inverse theorem for the Gowers norms. This paper studies these nilspaces per se, and in connection with further refinements of this inverse theorem that have been conjectured recently. Our first main result states that every compact finite-rank nilspace is obtained by taking a free nilspace (a nilspace based on an abelian group of the form $\mathbb{Z}^{r}\times \mathbb{R}^s$) and quotienting this by a discrete group action of a specific type, describable in terms of polynomials. We call these group actions "higher-order lattice actions", as they generalize actions of lattices in $\mathbb{Z}^r\times \mathbb{R}^s$. The second main result (which relies on the first one) represents every compact finite-rank nilspace as a double-coset space $K\backslash G / Γ$ where $G$ is a nilpotent Lie group of a specific kind. Our third main result extends the aforementioned results to $k$-step compact nilspaces (not necessarily of finite rank), by representing any such nilspace as a quotient of infinite products of free nilspaces and also as double coset spaces $K\backslash G/Γ$ where $G$ is a degree-$k$ nilpotent pro-Lie group. These results open the study of compact nilspaces to areas more classical than nilspace theory, such as the theory of topological group actions. The results also require developing the theory of topological non-compact nilspaces, for which we provide groundwork in this paper. Applications include new inverse theorems for Gowers norms on any finite abelian group. These theorems are purely group theoretic in that the correlating harmonics are based on double-coset spaces. This yields progress towards the Jamneshan-Tao conjecture.

math.CO↗

arXiv2025

MLPs at the EOC: Concentration of the NTK

We study the concentration of the Neural Tangent Kernel (NTK) $K_θ: \mathbb{R}^{m_0} \times \mathbb{R}^{m_0} \to \mathbb{R}^{m_l \times m_l}$ of $l$-layer Multilayer Perceptrons (MLPs) $N : \mathbb{R}^{m_0} \times Θ\to \mathbb{R}^{m_l}$ equipped with activation functions $ϕ(s) = a s + b \vert s \vert$ for some $a,b \in \mathbb{R}$ with the parameter $θ\in Θ$ being initialized at the Edge Of Chaos (EOC). Without relying on the gradient independence assumption that has only been shown to hold asymptotically in the infinitely wide limit, we prove that an approximate version of gradient independence holds at finite width. Showing that the NTK entries $K_θ(x_{i_1},x_{i_2})$ for $i_1,i_2 \in [1:n]$ over a dataset $\{x_1,\cdots,x_n\} \subset \mathbb{R}^{m_0}$ concentrate simultaneously via maximal inequalities, we prove that the NTK matrix $K(θ) = [\frac{1}{n} K_θ(x_{i_1},x_{i_2}) : i_1,i_2 \in [1:n]] \in \mathbb{R}^{nm_l \times nm_l}$ concentrates around its infinitely wide limit $\overset{\scriptscriptstyle\infty}{K} \in \mathbb{R}^{nm_l \times nm_l}$ without the need for linear overparameterization. Our results imply that in order to accurately approximate the limit, hidden layer widths have to grow quadratically as $m_k = k^2 m$ for some $m \in \mathbb{N}+1$ for sufficient concentration. For such MLPs, we obtain the concentration bound $\mathbb{P}( \Vert K(θ) - \overset{\scriptscriptstyle\infty}{K} \Vert \leq O((Δ_ϕ^{-2} + m_l^{\frac{1}{2}} l) κ_ϕ^2 m^{-\frac{1}{2}})) \geq 1-O(m^{-1})$ modulo logarithmic terms, where we denoted $Δ_ϕ= \frac{b^2}{a^2+b^2}$ and $κ_ϕ= \frac{\vert a \vert + \vert b \vert}{\sqrt{a^2 + b^2}}$. This reveals in particular that the absolute value ($Δ_ϕ=1$, $κ_ϕ=1$) beats the ReLU ($Δ_ϕ=\frac{1}{2}$, $κ_ϕ=\sqrt{2}$) in terms of the concentration of the NTK.

cs.LG↗

arXiv2025

MLPs at the EOC: Spectrum of the NTK

We study the properties of the Neural Tangent Kernel (NTK) $\overset{\scriptscriptstyle\infty}{K} : \mathbb{R}^{m_0} \times \mathbb{R}^{m_0} \to \mathbb{R}^{m_l \times m_l}$ corresponding to infinitely wide $l$-layer Multilayer Perceptrons (MLPs) taking inputs from $\mathbb{R}^{m_0}$ to outputs in $\mathbb{R}^{m_l}$ equipped with activation functions $ϕ(s) = a s + b \vert s \vert$ for some $a,b \in \mathbb{R}$ and initialized at the Edge Of Chaos (EOC). We find that the entries $\overset{\scriptscriptstyle\infty}{K}(x_1,x_2)$ can be approximated by the inverses of the cosine distances of the activations corresponding to $x_1$ and $x_2$ increasingly better as the depth $l$ increases. By quantifying these inverse cosine distances and the spectrum of the matrix containing them, we obtain tight spectral bounds for the NTK matrix $\overset{\scriptscriptstyle\infty}{K} = [\frac{1}{n} \overset{\scriptscriptstyle\infty}{K}(x_{i_1},x_{i_2}) : i_1, i_2 \in [1:n]]$ over a dataset $\{x_1,\cdots,x_n\} \subset \mathbb{R}^{m_0}$, transferred from the inverse cosine distance matrix via our approximation result. Our results show that $Δ_ϕ= \frac{b^2}{a^2+b^2}$ determines the rate at which the condition number of the NTK matrix converges to its limit as depth increases, implying in particular that the absolute value ($Δ_ϕ=1$) is better than the ReLU ($Δ_ϕ=\frac{1}{2}$) in this regard.

cs.LG↗

arXiv2025

Spectral algorithms in higher-order Fourier analysis

Our goal is to provide simple and practical algorithms in higher-order Fourier analysis which are based on spectral decompositions of operators. We propose a general framework for such algorithms and provide a detailed analysis of the quadratic case. Our results reveal new spectral aspects of the theory underlying higher-order Fourier analysis. Along these lines, we prove new inverse and regularity theorems for the Gowers norms based on higher-order character decompositions. Using these results, we prove a spectral inverse theorem and a spectral regularity theorem in quadratic Fourier analysis.

math.CO↗

arXiv2023

On the inverse theorem for Gowers norms in abelian groups of bounded torsion

In recent work, Jamneshan, Shalom and Tao proved an inverse theorem for the Gowers $U^{k+1}$-norm on finite abelian groups of fixed torsion $m$, where the final correlating harmonic is a polynomial phase function of degree at most $C(k,m)$. They also posed a related central question, namely, whether the bound $C$ can be reduced to the optimal value $k$ for every $m$. We make progress on this question using nilspace theory. First we connect the question to the study of finite nilspaces whose structure groups have torsion $m$. Then we prove one of the main results of this paper: a primary decomposition theorem for finite nilspaces, extending the Sylow decomposition in group theory. Thus we give an analogue for nilspaces of an ergodic-theoretic Sylow decomposition in the aforementioned work of Jamneshan-Shalom-Tao. We deduce various consequences which illustrate the following general idea: the primary decomposition enables a reduction of higher-order Fourier analysis in the $m$-torsion setting to the case of abelian $p$-groups. These consequences include a positive answer to the question of Jamneshan-Shalom-Tao when $m$ is squarefree, and also a new relation between uniformity norms and certain generalized cut norms on products of abelian groups of coprime orders. Another main result in this paper is a positive answer to the above central question for the $U^3$-norm, proving that $C(2,m)=2$ for all $m$. Finally, we give a partial answer to the question for all $k$ and $m$, proving an inverse theorem involving extensions of polynomial phase functions which were introduced by the third-named author, known as projected phase polynomials of degree $k$. A notable aspect is that this inverse theorem implies that of Jamneshan-Shalom-Tao, while involving projected phase polynomials of degree $k$, which are genuine obstructions to having small $U^{k+1}$-norm.

math.CO↗

arXiv2023

On measure-preserving $\mathbb{F}_p^ω$-systems of order $k$

Building on previous work in the nilspace-theoretic approach to the study of Host-Kra factors of measure-preserving systems, we prove that every ergodic $\mathbb{F}_p^ω$-system of order $k$ is a factor of an Abramov $\mathbb{F}_p^ω$-system of order $k$. This answers a question of Jamneshan, Shalom and Tao.

math.DS↗

arXiv2023

A Framework for Overparameterized Learning

A candidate explanation of the good empirical performance of deep neural networks is the implicit regularization effect of first order optimization methods. Inspired by this, we prove a convergence theorem for nonconvex composite optimization, and apply it to a general learning problem covering many machine learning applications, including supervised learning. We then present a deep multilayer perceptron model and prove that, when sufficiently wide, it $(i)$ leads to the convergence of gradient descent to a global optimum with a linear rate, $(ii)$ benefits from the implicit regularization effect of gradient descent, $(iii)$ is subject to novel bounds on the generalization error, $(iv)$ exhibits the lazy training phenomenon and $(v)$ enjoys learning rate transfer across different widths. The corresponding coefficients, such as the convergence rate, improve as width is further increased, and depend on the even order moments of the data generating distribution up to an order depending on the number of layers. The only non-mild assumption we make is the concentration of the smallest eigenvalue of the neural tangent kernel at initialization away from zero, which has been shown to hold for a number of less general models in contemporary works. We present empirical evidence supporting this assumption as well as our theoretical claims.

cs.LG↗

arXiv2023

Optimal transport with $f$-divergence regularization and generalized Sinkhorn algorithm

Entropic regularization provides a generalization of the original optimal transport problem. It introduces a penalty term defined by the Kullback-Leibler divergence, making the problem more tractable via the celebrated Sinkhorn algorithm. Replacing the Kullback-Leibler divergence with a general $f$-divergence leads to a natural generalization. The case of divergences defined by superlinear functions was recently studied by Di Marino and Gerolin. Using convex analysis, we extend the theory developed so far to include all $f$-divergences defined by functions of Legendre type, and prove that under some mild conditions, strong duality holds, optimums in both the primal and dual problems are attained, the generalization of the $c$-transform is well-defined, and we give sufficient conditions for the generalized Sinkhorn algorithm to converge to an optimal solution. We propose a practical algorithm for computing an approximate solution of the optimal transport problem with $f$-divergence regularization via the generalized Sinkhorn algorithm. Finally, we present experimental results on synthetic 2-dimensional data, demonstrating the effects of using different $f$-divergences for regularization, which influences convergence speed, numerical stability and sparsity of the optimal coupling.

math.OC↗

arXiv2022

On higher-order Fourier analysis in characteristic $p$

In 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 $\mathbb{F}_p^ω$-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 $k$-th Host-Kra factor is an Abramov system of order $\leq k$, 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 above-mentioned structure theorem yields a new proof of the Tao-Ziegler inverse theorem for Gowers norms on $\mathbb{F}_p^n$.

math.DS↗

arXiv2022

A refinement of Cauchy-Schwarz complexity

We introduce a notion of complexity for systems of linear forms, called sequential Cauchy-Schwarz complexity, which is parametrized by two positive integers $k,\ell$ and refines the notion of Cauchy-Schwarz complexity introduced by Green and Tao. We prove that if a system of linear forms has sequential Cauchy-Schwarz complexity at most $(k,\ell)$ then any average of 1-bounded functions over this system is controlled by the $2^{1-\ell}$-th power of the Gowers $U^{k+1}$-norms of the functions. For $\ell=1$ this agrees with Cauchy-Schwarz complexity, but for $\ell>1$ there are systems that have sequential Cauchy-Schwarz complexity at most $(k,\ell)$ whereas their Cauchy-Schwarz complexity is greater than $k$. Our main application illustrates this with systems over a prime field $\mathbb{F}_p$ that are denoted by $Φ_{k,M}$ and can be viewed as $M$-dimensional arithmetic progressions of length $k$. For each $M\geq 2$ we prove that $Φ_{k,M}$ has sequential Cauchy-Schwarz complexity at most $(k-2,|Φ_{k,M}|)$ (where $|Φ_{k,M}|$ is the number of forms in the system), whereas the Cauchy-Schwarz complexity of $Φ_{k,M}$ can be greater than $k-2$. Thus we obtain polynomial true-complexity bounds for $Φ_{k,M}$ with exponent $2^{-|Φ_{k,M}|}$. A recent general theorem of Manners, proved independently with different methods, implies a similar application but with different polynomial true-complexity bounds, as explained in the paper. In separate work, we use our application to give a new proof of the inverse theorem for Gowers norms on $\mathbb{F}_p^n$, and related results concerning ergodic actions of $\mathbb{F}_p^ω$.

math.CO↗

arXiv2022

On $\mathbb{F}_2^ω$-affine-exchangeable probability measures

For any standard Borel space $B$, let $\mathcal{P}(B)$ denote the space of Borel probability measures on $B$. In relation to a difficult problem of Aldous in exchangeability theory, and in connection with arithmetic combinatorics, Austin raised the question of describing the structure of affine-exchangeable probability measures on product spaces indexed by the vector space $\mathbb{F}_2^ω$, i.e., the measures in $\mathcal{P}(B^{\mathbb{F}_2^ω})$ that are invariant under the coordinate permutations on $B^{\mathbb{F}_2^ω}$ induced by all affine automorphisms of $\mathbb{F}_2^ω$. We answer this question by describing the extreme points of the space of such affine-exchangeable measures. We prove that there is a single structure underlying every such measure, namely, a random infinite-dimensional cube (sampled using Haar measure adapted to a specific filtration) on a group that is a countable power of the 2-adic integers. Indeed, every extreme affine-exchangeable measure in $\mathcal{P}(B^{\mathbb{F}_2^ω})$ is obtained from a $\mathcal{P}(B)$-valued function on this group, by a vertex-wise composition with this random cube. The consequences of this result include a description of the convex set of affine-exchangeable measures in $\mathcal{P}(B^{\mathbb{F}_2^ω})$ equipped with the vague topology (when $B$ is a compact metric space), showing that this convex set is a Bauer simplex. We also obtain a correspondence between affine-exchangeability and limits of convergent sequences of (compact-metric-space valued) functions on vector spaces $\mathbb{F}_2^n$ as $n\to\infty$. Via this correspondence, we establish the above-mentioned group as a general limit domain valid for any such sequence.

math.PR↗

arXiv2020

On sets with small sumset and m-sum-free sets in Z/pZ

The $3k-4$ conjecture in groups $\mathbb{Z}/p\mathbb{Z}$ for $p$ prime states that if $A$ is a nonempty subset of $\mathbb{Z}/p\mathbb{Z}$ satisfying $2A\neq \mathbb{Z}/p\mathbb{Z}$ and $|2A|=2|A|+r \leq \min\{3|A|-4,\;p-r-4\}$, then $A$ is covered by an arithmetic progression of size at most $|A|+r+1$. A theorem of Serra and Zémor proves the conjecture provided $r\leq 0.0001|A|$, without any additional constraint on $|A|$. Subject to the mild additional constraint $|2A|\leq 3p/4$ (which is optimal in a sense explained in the paper), our first main result improves the bound on $r$, allowing $r\leq 0.1368|A|$. We also prove a variant which further improves this bound on $r$ provided $A$ is sufficiently dense. We then give several applications. First we apply the above variant to give a new upper bound for the maximal density of $m$-sum-free sets in $\mathbb{Z}/p\mathbb{Z}$, i.e., sets $A$ having no solution $(x,y,z)\in A^3$ to the equation $x+y=mz$, where $m\geq 3$ is a fixed integer. The previous best upper bound for this maximal density was $1/3.0001$ (using the Serra-Zémor Theorem). We improve this to $1/3.1955$. We also present a construction following an idea of Schoen, which yields a lower bound for this maximal density of the form $1/8+o(1)_{p\to\infty}$. Another application of our main results concerns sets of the form $\frac{A+A}{A}$ in $\mathbb{F}_p$, and we also improve the structural description of large sum-free sets in $\mathbb{Z}/p\mathbb{Z}$.

math.CO↗

arXiv2020

New estimates for exponential sums over multiplicative subgroups and intervals in prime fields

Let ${\mathcal H}$ be a multiplicative subgroup of $\mathbb{F}_p^*$ of order $H>p^{1/4}$. We show that $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}} {\mathbf{\,e}}_p(ax)\right| \le H^{1-31/2880+o(1)}, $$ where ${\mathbf{\,e}}_p(z) = \exp(2 πi z/p)$, which improves a result of Bourgain and Garaev (2009). We also obtain new estimates for double exponential sums with product $nx$ with $x \in {\mathcal H}$ and $n \in {\mathcal N}$ for a short interval ${\mathcal N}$ of consecutive integers.

math.NT↗

arXiv2019

On nilspace systems and their morphisms

A nilspace system is a generalization of a nilsystem, consisting of a compact nilspace X equipped with a group of nilspace translations acting on X. Nilspace systems appear in different guises in several recent works, and this motivates the study of these systems per se as well as their relation to more classical types of systems. In this paper we study morphisms of nilspace systems, i.e., nilspace morphisms with the additional property of being consistent with the actions of the given translations. A nilspace morphism does not necessarily have this property, but one of our main results shows that it factors through some other morphism which does have the property. As an application we obtain a strengthening of the inverse limit theorem for compact nilspaces, valid for nilspace systems. This is used in work of the first and third named authors to generalize the celebrated structure theorem of Host and Kra on characteristic factors.

math.DS↗

arXiv2018

A note on the Bilinear Bogolyubov Theorem: Transverse and bilinear sets

A set $P\subset \mathbb{F}_p^n\times\mathbb{F}_p^n$ is called $\textit{bilinear}$ when it is the zero set of a family of linear and bilinear forms, and $\textit{transverse}$ when it is stable under vertical and horizontal sums. A theorem of the first author provides a generalization of Bogolyubov's theorem to the bilinear setting. Roughly speaking, it implies that any dense transverse set $P\subset \mathbb{F}_p^n\times\mathbb{F}_p^n$ contains a large bilinear set. In this paper, we elucidate the extent to which a transverse set is forced to be (and not only contain) a bilinear set.

math.CO↗

arXiv2018

A Plünnecke-Ruzsa inequality in compact abelian groups

The Plünnecke-Ruzsa inequality is a fundamental tool to control the growth of finite subsets of abelian groups under repeated addition and subtraction. Other tools to handle sumsets have gained applicability by being extended to more general subsets of more general groups. This motivates extending the Plünnecke-Ruzsa inequality, in particular to measurable subsets of compact abelian groups by replacing the cardinality with the Haar probability measure. This objective is related to the question of the stability of classes of Haar measurable sets under addition. In this direction the class of analytic sets is a natural one to work with. We prove a Plünnecke-Ruzsa inequality for K-analytic sets in general compact (Hausdorff) abelian groups. We also discuss further extensions, some of which raise questions of independent interest in descriptive topology.

math.CO↗