arXiv subjects

Carlos Pastor

Publications and source records attributed to Carlos Pastor.

4 recordsLinked to original sources

arXiv2017

On the regularity of fractional integrals of modular forms

In this paper we study some local and global regularity properties of Fourier series obtained as fractional integrals of modular forms. In particular we characterize the differentiability at rational points, determine their Hölder exponent everywhere (using several definitions) and compute the associated spectrum of singularities. We also show that these functions satisfy an approximate functional equation, and use it to discuss the graphs of "Riemann's example" and of fractional integrals of cusp forms for $Γ_0(N)$. We include some computer plots.

math.CA↗

arXiv2017

Lattice points in elliptic paraboloids

We consider the lattice point problem corresponding to a family of elliptic paraboloids in $\mathbb{R}^d$ with $d\ge3$ and we prove the expected to be optimal exponent, improving previous results. This is especially noticeable for $d=3$ because the optimal exponent is conjectural even for the sphere. We also treat some aspects of the case $d=2$, getting for a simple parabolic region an $Ω$-result that is unknown for the classical circle and divisor problems.

math.NT↗

arXiv2017

Some more counterexamples for Bombieri's conjecture on univalent functions

We disprove a conjecture of Bombieri regarding univalent functions in the unit disk in some previously unknown cases. The key step in the argument is showing that the global minimum of the real function $\big(n\sin{x}-\sin(nx)\big)/\big(m\sin{x}-\sin(mx)\big)$ is attained at $x = 0$ for integers $m>n\geq2$ when $m$ is odd and $n$ is even, $m$ is sufficiently big and $0.5 \leq n/m \leq 0.8194$.

math.CV↗

arXiv2017

Lattice points in bodies of revolution II

In a previous article it was shown that when a three-dimensional smooth convex body has rotational symmetry around a coordinate axis one can find better bounds for the lattice point discrepancy than what is known for more general convex bodies. To accomplish this, however, it was necessary to assume a non-vanishing condition on the third derivative of the generatrix. In this article we drop this condition, showing that the aforementioned bound holds for a wider family of revolution bodies, which includes those with analytic boundary. A novelty in our approach is that, besides the usual analytic methods, it requires studying some Diophantine properties of the Taylor coefficients of the phase on the Fourier transform side.

math.NT↗