arXiv subjects

Adrián Ubis

Publications and source records attributed to Adrián Ubis.

7 recordsLinked to original sources

arXiv2020

Effective equidistribution of translates of large submanifolds in semisimple homogeneous spaces

Let $G=SL_2(\mathbb R)^d$ and $Γ=Γ_0^d$ with $Γ_0$ a lattice in $SL_2(\mathbb R)$. Let $S$ be any "curved" submanifold of small codimension of a maximal horospherical subgroup of $G$ relative to an $\mathbb R$-diagonalizable element $a$ in the diagonal of $G$. Then for $S$ compact our result can be described by saying that $a^n \text{vol}_S$ converges in an effective way to the volume measure of $G/Γ$ when $n\to \infty$, with $\text{vol}_S$ the volume measure on $S$.

math.DS↗

arXiv2019

Primos, paridad y análisis

Distinguir entre enteros con un número par o impar de divisores primos es una de las tareas más difíciles en la teoría analítica de números. Un trabajo reciente de Matomäki y Radziwiłł muestra que, en promedio, ambos existen con la misma frecuencia aún en intervalos muy cortos. Este avance ya ha tenido varias aplicaciones importantes en las manos de Matomäki, Radziwiłł, Tao y Teräväinen. Explicaremos en detalle una prueba completa del resultado original de Matomäki y Radziwiłł, así como de varias aplicaciones. ----- To distinguish between integers with an even or an odd number of prime factors is one of the most difficult tasks in Analytic Number Theory. A recent work by Matomäki and Radziwiłł shows that, in average, both types of integers appear with the same frequency even in very short intervals. This breakthrough has already had several applications in the hands of Matomäki, Radziwiłł, Tao and Teräväinen. We explain in detail the complete proof of both the original result by Matomäki and Radziwiłł and of some of its applications.

math.NT↗

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

Invariant subspaces for Bishop operators and beyond

Bishop operators $T_α$ acting on $L^2[0,1)$ were proposed by E. Bishop in the fifties as possible operators which might entail counterexamples for the Invariant Subspace Problem. We prove that all the Bishop operators are biquasitriangular and, derive as a consequence that they are norm limits of nilpotent operators. Moreover, by means of arithmetical techniques along with a theorem of Atzmon, the set of irrationals $α\in (0,1)$ for which $T_α$ is known to possess non-trivial closed invariant subspaces is considerably enlarged, extending previous results by Davie, MacDonald and Flattot. Furthermore, we essentially show that when our approach fails to produce invariant subspaces it is actually because Atzmon Theorem cannot be applied. Finally, upon applying arithmetical bounds obtained, we deduce local spectral properties of Bishop operators proving, in particular, that neither of them satisfy the Dunford property $(C)$.

math.FA↗

arXiv2018

Integral points on convex curves

We estimate the maximal number of integral points which can be on a convex arc in the plane with given length, minimal radius of curvature and initial slope.

math.NT↗

arXiv2014

Local $L^2$-regularity of Riemann's Fourier series

We are interested in the convergence and the local regularity of the lacunary Fourier series $F_s(x) = \sum_{n=1}^{+\infty} \frac{e^{2iπn^2 x}}{n^s}$. In the 1850's, Riemann introduced the series $F_2$ as a possible example of nowhere differentiable function, and the study of this function has drawn the interest of many mathematicians since then. We focus on the case when $1/2<s\leq 1$, and we prove that $F_s(x)$ converges when $x$ satisfies a Diophantine condition. We also study the $L^2$- local regularity of $F_s$, proving that the local $L^2$-norm of $F_s$ around a point $x$ behave differently around different $x$, according again to Diophantine conditions on $x$.

math.FA↗

arXiv2012

Multifractal behavior of polynomial Fourier series

We prove non-trivial upper and lower bounds for the "Spectrum of Singularities" of Fourier Series with polynomial frequencies. The Spectrum of Singularities of a function f gives the Hausdorff dimension of the set of points with a given Hölder exponent for f.

math.NT↗