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Antonio Córdoba

Publications and source records attributed to Antonio Córdoba.

11 recordsLinked to original sources
arXiv2026

Curly Kakeya: The Hausdorff dimension of sets containing circles or line segments in many directions

In this paper, we establish a lower bound for the Hausdorff dimension of a set $K\subset \mathbb{R}^n$ that contains a dilated and translated copy of every meridian, that is, of every $(d-2)$-dimensional sphere passing through the poles of $S^{d-1}$, a $(d-1)$-dimensional sphere, with $3\le d\le n$, and $S^{d-1}$ contained in $\mathbb{R}^n$. Under this assumption, we have \[ \dim_H(K)\ge d-1. \] This result is closely related to, and reminiscent of, the classical Kakeya set problem. We build upon this connection, employing similar techniques to show that if $K\subset \mathbb{R}^n$ contains a unit straight line segment in every direction corresponding to a smooth curve on $S^{n-1}$, then its Hausdorff dimension is $\ge 2$.

math.MG↗
arXiv2025

On sets with missing differences in compact abelian groups

A much-studied problem posed by Motzkin asks to determine, given a finite set $D$ of integers, the so-called Motzkin density for $D$, i.e., the supremum of upper densities of sets of integers whose difference set avoids $D$. We study the natural analogue of this problem in compact abelian groups. Using ergodic-theoretic tools, this is shown to be equivalent to the following discrete problem: given a lattice $Λ\subset \mathbb{Z}^r$, letting $D$ be the image in $\mathbb{Z}^r/Λ$ of the standard basis, determine the Motzkin density for $D$ in $\mathbb{Z}^r/Λ$. We study in particular the periodicity question: is there a periodic $D$-avoiding set of maximal density in $\mathbb{Z}^r/Λ$? The Greenfeld--Tao counterexample to the periodic tiling conjecture implies that the answer can be negative. On the other hand, we prove that the answer is positive in several cases, including the case rank$(Λ)=1$ (in which we give a formula for the Motzkin density), the case rank$(Λ)=r-1$, and hence also the case $r\leq 3$. It follows that, for up to three missing differences, the Motzkin density in a compact abelian group is always a rational number.

math.CO↗
arXiv2018

Fourier series in BMO with number theoretical implications

We introduce an elementary argument to bound the $\textrm{BMO}$ seminorm of Fourier series with gaps giving in particular a sufficient condition for them to be in this space. Using finer techniques we carry out a detailed study of the series $\sum n^{-1}e^{2πi n^2 x}$ providing some insight into how much this $\text{BMO}$ Fourier series differs from defining an $L^\infty$ function.

math.CA↗
arXiv2018

A conjecture of Antoni Zygmund

In this paper we establish an exponential covering theorem implying a conjecture formulated by A. Zygmund circa 1935 whose three-dimensional case was obtained by the first named author in 1978.

math.CA↗
arXiv2017

Uniqueness for SQG patch solutions

This paper is about the evolution of a temperature front governed by the Surface quasi-geostrophic equation. The existence part of that program within the scale of Sobolev spaces was obtained by one of the authors [10]. Here we revisit that proof introducing some new tools and points of view which allow us to conclude the also needed uniqueness result.

math.AP↗
arXiv2016

Radial multipliers and restriction to surfaces of the Fourier transform in mixed-norm spaces

In this article we revisit some classical conjectures in harmonic analysis in the setting of mixed norm spaces $L^p_{rad} L^2_{ang} (\mathbb{R}^n)$. We produce sharp bounds for the restriction of the Fourier transform to compact hypersurfaces of revolution in the mixed norm setting and study an extension of the disc multiplier. We also present some results for the discrete restriction conjecture and state an intriguing open problem.

math.CA↗
arXiv2014

Core regulatory network motif underlies the ocellar complex patterning in Drosophila melanogaster

During organogenesis, developmental programs governed by Gene Regulatory Networks (GRN) define the functionality, size and shape of the different constituents of living organisms. Robustness, thus, is an essential characteristic that GRNs need to fulfill in order to maintain viability and reproducibility in a species. In the present work we analyze the robustness of the patterning for the ocellar complex formation in Drosophila melanogaster fly. We have systematically pruned the GRN that drives the development of this visual system to obtain the minimum pathway able to satisfy this pattern. We found that the mechanism underlying the patterning obeys to the dynamics of a 3-nodes network motif with a double negative feedback loop fed by a morphogenetic gradient that triggers the inhibition in a French flag problem fashion. A Boolean modeling of the GRN confirms robustness in the patterning mechanism showing the same result for different network complexity levels. Interestingly, the network provides a steady state solution in the interocellar part of the patterning and an oscillatory regime in the ocelli. This theoretical result predicts that the ocellar pattern may underlie oscillatory dynamics in its genetic regulation.

q-bio.TO↗
arXiv2014

Collective computation in a network with distributed information

We analyze a distributed information network in which each node has access to the information contained in a limited set of nodes (its neighborhood) at a given time. A collective computation is carried out in which each node calculates a value that implies all information contained in the network (in our case, the average value of a variable that can take different values in each network node). The neighborhoods can change dynamically by exchanging neighbors with other nodes. The results of this collective calculation show rapid convergence and good scalability with the network size. These results are compared with those of a fixed network arranged as a square lattice, in which the number of rounds to achieve a given accuracy is very high when the size of the network increases. The results for the evolving networks are interpreted in light of the properties of complex networks and are directly relevant to the diameter and characteristic path length of the networks, which seem to express "small world" properties.

cs.SI↗
arXiv2013

On Conic Fourier Multipliers

We prove a weighted inequality which controls conic Fourier multiplier operators in terms of lacunary directional maximal operators. By bounding the maximal operators, this enables us to conclude that the multiplier operators are bounded on $L^p(\mathbb{R}^3)$ with $1<p<\infty$.

math.CA↗