arXiv subjects

Cynthia Bortolotto

Publications and source records attributed to Cynthia Bortolotto.

6 recordsLinked to original sources

arXiv2026

A complex-analytic proof of square-restricted stable phase retrieval in Fock space

We give a short complex-analytic proof of a square-restricted form of local stable phase retrieval at the Gaussian in one-dimensional Fock space. The main estimate is a coercivity inequality for the map $F\mapsto F^2$: \[ \|F^2-F(0)^2|_{\mathcal{F}^2(\mathbb{C})} \lesssim \inf_{c\in\mathbb{R}}\||F|^2-c\|_{L^2(d γ)}. \] The proof uses a weighted derivative norm, two integrations by parts, and a weighted Cauchy inequality. The proof has been completely verified in Lean with the aid of Large Language Models.

math.CV↗

arXiv2026

Modular Forms and Numerical Explorations of Rational Approximations to $ζ(3)$

We revisit Beukers' modular-form proof of the irrationality of $ζ(3)$ from the point of view of the auxiliary weight two modular form. For the Fricke group $Γ_0(6)^\star$, we show that Beukers' choice is not isolated: it belongs to a one-parameter affine family. These approximations have the same exponential decay as the classical Apéry approximations and satisfy the same denominator-growth estimate needed in Beukers' irrationality argument. We then apply the same construction to several other genus-zero Fricke groups.

math.NT↗

arXiv2025

Intersections of random chords of a circle

Where are the intersection points of diagonals of a regular $n$-gon located? What is the distribution of the intersection point of two random chords of a circle? We investigate these and related new questions in geometric probability, extend a largely forgotten result of Karamata, and elucidate its connection to the Bertrand paradox.

math.MG↗

arXiv2024

On Extremal Problems Associated with Random Chords on a Circle

Inspired by the work of Karamata, we consider an extremization problem associated with the probability of intersecting two random chords inside a circle of radius $r, \, r \in (0,1]$, where the endpoints of the chords are drawn according to a given probability distribution on $\mathbb{S}^1$. We show that, for $r=1,$ the problem is degenerated in the sense that any continuous measure is an extremiser, and that, for $r$ sufficiently close to $1,$ the desired maximal value is strictly below the one for $r=1$ by a polynomial factor in $1-r.$ Finally, we prove, by considering the auxiliary problem of drawing a single random chord, that the desired maximum is $1/4$ for $r \in (0,1/2).$ Connections with other variational problems and energy minimization problems are also presented.

math.MG↗

arXiv2023

Weyl sums with multiplicative coefficients and joint equidistribution

In this paper we generalize a result of Montgomery and Vaughan regarding exponential sums with multiplicative coefcients to the setting of Weyl sums. As applications, we establish a joint equidistribution result for roots of polynomial congruences and polynomial values and obtain some new results for mixed character sums.

math.NT↗

arXiv2020

On the endpoint behaviour of oscillatory maximal function

Inspired by a question of Lie, we study boundedness in subspaces of $L^1(\mathbb{R})$ of oscillatory maximal functions. In particular, we construct functions in $L^1(\mathbb{R})$ which are never integrable under action of our class of maximal functions. On the other hand, we prove that these maximal functions map certain classes of spaces resembling Sobolev spaces into $L^1(\mathbb{R})$ continuously under mild curvature assumptions on the phase $γ$.

math.CA↗