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Ujué Etayo

Publications and source records attributed to Ujué Etayo.

13 recordsLinked to original sources
arXiv2026

Discrepancy of determinantal point processes on compact, connected two-point homogeneous spaces

We study the $L^{\infty}$ discrepancy of point sets generated by determinantal point processes on all compact, connected two-point homogeneous spaces, namely spheres and projective spaces. Using concentration inequalities and variance estimates for the number of points in metric balls, we derive general upper bounds for the discrepancy of homogeneous determinantal point processes. In the particular case of the harmonic ensemble, we show that the discrepancy of $N$ points is $O((N^{1-1/D})^{1/2}\log N)$ with high probability, where $D$ denotes the real dimension of the manifold. For the projective ensemble on $\mathbb{CP}^d$, we obtain the sharper bound $O((N^{1-1/D}\log N)^{1/2})$. These results extend previously known discrepancy estimates for determinantal point processes on the sphere to all compact, connected two-point homogeneous spaces.

math.CA↗
arXiv2026

Random points on $\mathbb{S}^3$ with small logarithmic energy

We analyse several constructions of random point sets on the sphere $\mathbb{S}^{3}\subset\mathbb{R}^4$ evaluating and comparing them through their discrete logarithmic energy: \begin{equation*} E_0(ω_N) = \sum_{\substack{i, j=1\\ i \neq j}}^{N} \log\frac{1}{\|x_i - x_j\|}, \; \text{ where}\; ω_N=\{x_1,\ldots,x_N\} \subset \mathbb{S}^3. \end{equation*} Using the Hopf fibration, we lift a range of well-distributed families of points from the $2$-dimensional sphere - including uniformly random points, antipodally symmetric sets, determinantal point processes, and the Diamond ensemble - to $\mathbb{S}^{3}$, in order to assess their energy performance. In particular, we carry out this asymptotic analysis for the Spherical ensemble (a well known determinantal point process on $\mathbb{S}^2$), obtaining as a result a family of points on the $3$-dimensional sphere whose logarithmic energy is asymptotically the lowest achieved to date. This, in turn, provides a new upper bound for the minimal logarithmic energy on $\mathbb{S}^3$. Although an analytic treatment of the lifted Diamond ensemble remains elusive, extensive simulations presented here show that its empirical energies lie below all other deterministic and non-deterministic constructions considered. Together, these results sharpen the quantitative link between potential-theoretic optima on $\mathbb{S}^{2}$ and $\mathbb{S}^{3}$ and provide both theoretical and numerical benchmarks for future work.

math.PR↗
arXiv2025

A Bombieri-type inequality and equidistribution of points

In recent work, Etayo introduces a new Bombieri-type inequality for monic polynomials. Here we reinterpret this new inequality as a more general integral inequality involving the Green function for the sphere. This rather geometric interpretation allows for generalizations of the basic inequality, involving fractional zeros while also opening up the possibility to extend the setting to general compact Riemann surfaces. We derive a sharp form of these generalized Bombieri-type inequalities for the case of the sphere and the torus. These inequalities involve a quantity we call the packing number, which in turn is inspired by the geometric zero packing problems considered by Hedenmalm in the context of the asymptotic variance of the Bergman projection of a bounded function. As for the torus, we introduce analogs of polynomials (pseudopolynomials) based on the classical Weierstrass $σ$ function, and we explain how such pseudopolynomials fit in with the extended geometric Bombieri-type inequality. The sharpness of the packing number bound on the torus involves the construction of a lattice configuration on the torus for any given integer number of points. The corresponding bound for the sphere instead relies on the existence of well-conditioned polynomials in the sense of Shub and Smale.

math.CA↗
arXiv2024

Minimal Riesz and logarithmic energies on the Grassmannian $\operatorname{Gr}_{2,4}$

We study the Riesz and logarithmic energies on the Grassmannian $\operatorname{Gr}_{2,4}$ of $2$-dimensional subspaces of $\mathbb{R}^4$. We prove that the continuous Riesz and logarithmic energies are uniquely minimized by the uniform measure, and we obtain asymptotic upper and lower bounds for the minimal discrete energies, with matching orders for the next-order terms. Additionally, we define a determinantal point process on $\operatorname{Gr}_{2,4}$ and compute the expected energy of the points coming from this random process, thereby obtaining explicit constants in the upper bounds for the Riesz and logarithmic energies.

math.CA↗
arXiv2022

Low energy points on the sphere and the real projective plane

We present a generalization of a family of points on $\mathbb{S}^2$, the Diamond ensemble, containing collections of $N$ points on $\mathbb{S}^2$ with very small logarithmic energy for all $N\in\mathbb{N}$. We extend this construction to the real projective plane $\mathbb{RP}^2$ and we obtain upper and lower bounds with explicit constants for the Green and logarithmic energy on this last space.

math.CA↗
arXiv2019

A sharp Bombieri inequality, logarithmic energy and well conditioned polynomials

In this paper we explore the connections between minimizers of the discrete logarithmic energy on the 2-dimensional sphere, univariate polynomials with optimal condition number in the Shub-Smale sense and a quotient involving norms of polynomials. Our main results are that polynomials with optimal condition number produce spherical points with small logarithmic energy (the reverse result was already known) and a sharp Bombieri type inequality for univariate polynomials with complex coefficients.

math.CA↗
arXiv2019

Spherical cap discrepancy of the Diamond ensemble

We compute the spherical cap discrepancy of the Diamond ensemble (a set of evenly distributed spherical points) as well as some other quantities. We also define an area regular partition on the sphere where each region contains exactly one point of the Diamond ensemble. For a concrete choice of parameters, we prove that the Diamond ensemble provides the best spherical cap discrepancy known until date for a deterministic family of points.

math.NA↗
arXiv2018

The Diamond ensemble: a constructive set of points with small logarithmic energy

We define a family of random sets of points, the Diamond ensemble, on the sphere $\mathbb{S}^{2}$ depending on several parameters. Its most important property is that, for some of these parameters, the asymptotic expected value of the logarithmic energy of the points can be computed rigorously and shown to attain very small values, quite close to the conjectured minimal value.

math-ph↗
arXiv2018

A generalization of the spherical ensemble to even-dimensional spheres

In a recent article, Alishahi and Zamani discuss the spherical ensemble, a rotationally invariant determinantal point process on the 2-sphere. In this paper we extend this process in a natural way to the 2d-dimensional sphere. We prove that the expected value of the Riesz s-energy associated to this determinantal point process has a reasonably low value compared to the known asymptotic expansion of the minimal Riesz s-energy.

math.PR↗
arXiv2017

The projective ensemble and distribution of points in odd-dimensional spheres

We define a determinantal point process on the complex projective space that reduces to the so-called spherical ensemble for complex dimension 1 under identification of the 2-sphere with the Riemann sphere. Through this determinantal point process we propose a point processs in odd-dimensional spheres that produces fairly well-distributed points, in the sense that the expected value of the Riesz 2-energy for these collections of points is smaller than all previously known bounds.

math.CA↗
arXiv2016

Asymptotically optimal designs on compact algebraic manifolds

We find t-designs on compact algebraic manifolds with a number of points comparable to the dimension of the space of polynomials of degree t on the manifold. This generalizes results on the sphere by Bondarenko, Radchenko and Viazovska. Of special interest is the particular case of the Grassmannians where our results improve the bounds that had been proved previously.

math.CA↗