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Luca Brandolini

Publications and source records attributed to Luca Brandolini.

At least 19 recordsLinked to original sources
arXiv2026

Jittered sampling and probability measures

This paper investigates the discrepancy of a family of random sampling methods obtained by perturbing the grid $\frac{1}{M}\mathbb{Z}^{d}\cap\left[ -1/2,1/2\right)^{d}$, where $M$ is a large positive integer. Parameterized by an arbitrary probability measure $\mu$, this family encompasses several classical methods for evaluating the quality of an $N$-point set in $\mathbb{T}^{d}$, where $N=M^{d}$. We show that all probability measures, except for Dirac measures, behave like the Lebesgue measure in the Monte Carlo discrepancy. This represents a limiting case where the measure $\mu$ depends on $M$. In this latter context, we prove that, up to a constant, the lowest possible discrepancy is achieved when the support of $\mu$ has diameter $\leq c/M$, and that this upper bound is sharp.

math.NT↗
arXiv2026

Discrepancy estimates for multi-dimensional non-smooth convex bodies: a case study

We study $L^2$-averaged discrepancies of finite sequences of points in the torus $\mathbb{T}^d$ with respect to translated and dilated copies of convex bodies with non-smooth boundary. Under suitable anisotropic assumptions on the decay of the Fourier transform of the body, we prove matching lower and upper bounds for the averaged discrepancy, obtaining the rate $ N^{1 - \frac{d+1}{d^2+d-1}}$. This yields an intermediate regime between smooth convex bodies and polytopes and recovers the known exponent $2/5$ in dimension $d=2$. The argument relies on harmonic analysis techniques combined with averaging procedures adapted to the anisotropic setting. As an application, we analyze a class of convex bodies exhibiting mixed geometric features, including flat regions, curved parts, and edges.

math.CA↗
arXiv2026

Quadratic discrepancy estimates for probability measures on the Heisenberg group

We initiate the study of quadratic discrepancy for finite point sets on the Heisenberg group $\mathbb H^n$ with respect to upper Ahlfors regular probability measures. For a natural family of test sets given by left translations and dilations of cylindrically defined neighborhoods, we introduce an $L^2$-discrepancy and establish a Roth-type lower bound depending on the homogeneous dimension of $\mathbb H^n$. This result extends classical discrepancy estimates from the Euclidean and compact settings to a non-commutative, step-two nilpotent Lie group. It should be viewed as a first step toward the development of a discrepancy theory on the Heisenberg group.

math.CA↗
arXiv2025

Irregularities of distribution and Fourier transforms of multi-dimensional convex bodies

W. Schmidt, H. Montgomery, and J. Beck proved a result on irregularities of distribution with respect to $d$-dimensional balls. In this paper, we extend their result to any $d$-dimensional convex body with a smooth boundary and finite order of contact. As an intermediate step, we prove a geometric inequality for the Fourier transform of the characteristic function of a convex body.

math.NT↗
arXiv2024

Single radius spherical cap discrepancy on compact two-point homogeneous spaces

In this note we study estimates from below of the single radius spherical discrepancy in the setting of compact two-point homogeneous spaces. Namely, given a $d$-dimensional manifold $\mathcal M$ endowed with a distance $\rho$ so that $(\mathcal M, \rho)$ is a two-point homogeneous space and with the Riemannian measure $\mu$, we provide conditions on $r$ such that if $D_r$ denotes the discrepancy of the ball of radius $r$, then, for an absolute constant $C>0$ and for every set of points $\{x_j\}_{j=1}^N$, one has $\int_{\mathcal M} |D_{r}(x)|^2\, d\mu(x)\geqslant C N^{-1-\frac1d}$. The conditions on $r$ that we have depend on the dimension $d$ of the manifold and cannot be achieved when $d \equiv 1 \ ( \operatorname{mod}4)$. Nonetheless, we prove a weaker estimate for such dimensions as well.

math.CA↗
arXiv2023

On a discrete approach to lower bounds in discrepancy theory

In this paper, we prove that some renowned lower bounds in discrepancy theory admit a discrete analogue. Namely, we prove that the lower bound of the discrepancy for corners in the unit cube due to Roth holds true also for a suitable finite family of corners. We also prove two analogous results for the discrepancy on the torus with respect to squares and balls.

math.CA↗
arXiv2022

Irregularities of distribution on two point homogeneous spaces

We study the irregularities of distribution on two-point homogeneous spaces. Our main result is the following: let $d$ be the real dimension of a two point homogeneous space $\mathcal{M}$, let $\left( \{ a_{j}\} _{j=1}^{N},\{ x_{j}\} _{j=1}^{N}\right) $ be a system of positive weights and points on $\mathcal{M}$ and let \[ D_{r}( x) =\sum_{j=1}^{N}a_{j}\chi_{B_{r}(x)}(x_{j})-\mu(B_{r}(x)) \] be the discrepancy associated with the ball $B_{r}( x) $. Then, if $d\not \equiv 1(\operatorname{mod}4)$, for any radius $0<r<\pi/2$, we obtain the sharp estimate \[ \int_{\mathcal{M}}\left( \left\vert D_{r}( x) \right\vert ^{2}+\left\vert D_{2r}( x) \right\vert ^{2}\right) d\mu( x) \geqslant cN^{-1-\frac{1}{d}}. \]

math.AP↗
arXiv2022

Euler-MacLaurin summation formula on polytopes and expansions in multivariate Bernoulli polynomials

We provide a multidimensional weighted Euler--MacLaurin summation formula on polytopes and a multidimensional generalization of a result due to L. J. Mordell on the series expansion in Bernoulli polynomials. These results are consequences of a more general series expansion; namely, if $\chi _{\tau\mathcal{P}}$ denotes the characteristic function of a dilated integer convex polytope $\mathcal{P}$ and $q$ is a function with suitable regularity, we prove that the periodization of $q\chi_{\tau\mathcal{P}}$ admits an expansion in terms of multivariate Bernoulli polynomials. These multivariate polynomials are related to the Lerch Zeta function. In order to prove our results we need to carefully study the asymptotic expansion of $\widehat{q\chi_{\tau\mathcal{P}}}$, the Fourier transform of $q\chi _{\tau\mathcal{P}}$.

math.CA↗
arXiv2021

Irregularities of distribution and geometry of planar convex sets

We consider a planar convex body $C$ and we prove several analogs of Roth's theorem on irregularities of distribution. When $\partial C$ is $\mathcal{C}% ^{2}$ regardless of curvature, we prove that for every set $\mathcal{P}_{N}$ of $N$ points in $\mathbb{T}^{2}$ we have the sharp bound \[ \int_{0}^{1}\int_{\mathbb{T}^{2}}\left\vert \mathrm{card}\left( \mathcal{P}_{N}\mathcal{\cap}\left( \lambda C+t\right) \right) -\lambda ^{2}N\left\vert C\right\vert \right\vert ^{2}~dtd\lambda\geqslant cN^{1/2}\;. \] When $\partial C$ is only piecewise $\mathcal{C}^{2}$ and is not a polygon we prove the sharp bound% \[ \int_{0}^{1}\int_{\mathbb{T}^{2}}\left\vert \mathrm{card}\left( \mathcal{P}_{N}\mathcal{\cap}\left( \lambda C+t\right) \right) -\lambda ^{2}N\left\vert C\right\vert \right\vert ^{2}~dtd\lambda\geqslant cN^{2/5}. \] We also give a whole range of intermediate sharp results between $N^{2/5}$ and $N^{1/2}$. Our proofs depend on a lemma of Cassels-Montgomery, on ad hoc constructions of finite point sets, and on a geometric type estimate for the average decay of the Fourier transform of the characteristic function of $C$.

math.MG↗
arXiv2020

An Euler-Maclaurin formula for polygonal sums

We prove an Euler-Maclaurin formula for double polygonal sums and, as a corollary, we obtain approximate quadrature formulas for integrals of smooth functions over polygons with integer vertices. Our Euler-Maclaurin formula is in the spirit of Pick's theorem on the number of integer points in an integer polygon and involves weighted Riemann sums, using tools from Harmonic analysis. Finally, we also exhibit a classical trick, dating back to Huygens and Newton, to accelerate convergence of these Riemann sums.

math.CA↗
arXiv2020

On a sharp lemma of Cassels and Montgomery on manifolds

Let $\left( \mathcal{M},g\right) $ be a $d$-dimensional compact connected Riemannian manifold and let $\left\{ φ_{m}\right\}_{m=0}^{+\infty}$ be a complete sequence of orthonormal eigenfunctions of the Laplace-Beltrami operator on $\mathcal{M}$. We show that there exists a positive constant $C$ such that for all integers $N$ and $X$ and for all finite sequences of $N$ points in $\mathcal{M}$, $\left\{ x\left( j\right) \right\}_{j=1}^{N}$, and positive weights $\left\{ a_{j}\right\}_{j=1}^{N}$ we have \[ \sum_{m=0}^{X} | \sum_{j=1}^{N} a_{j} φ_{m} ( x( j) ) | ^{2}\geq \max \{ CX\sum_{j=1}^{N}a_{j}^{2},( \sum_{j=1}^{N}a_{j}) ^{2}\}.\]

math.AP↗
arXiv2019

Convergence of multiple Fourier series and Pick's theorem

We add another brick to the large building comprising proofs of Pick's theorem. Although our proof is not the most elementary, it is short and reveals a connection between Pick's theorem and the pointwise convergence of multiple Fourier series of piecewise smooth functions.

math.NT↗
arXiv2019

Fourier analytic techniques for lattice point discrepancy

Counting integer points in large convex bodies with smooth boundaries containing isolated flat points is oftentimes an intermediate case between balls (or convex bodies with smooth boundaries having everywhere positive curvature) and cubes (or convex polytopes). In this paper we provide a detailed description of several discrepancy problems in the particular planar case where the boundary coincides locally with the graph of the function $\mathbb{R\ni}t\mapsto\left\vert t\right\vert ^γ$, with $γ>2$. We consider both \textit{integer points} problems and \textit{irregularities of distribution} problems. The above \textquotedblleft restriction\textquotedblright\ to a particular family of convex bodies is compensated by the fact that many proofs are elementary. The paper is entirely self-contained.

math.FA↗
arXiv2018

Discrepancy for convex bodies with isolated flat points

We consider the discrepancy of the integer lattice with respect to the collection of all translated copies of a dilated convex body having a finite number of flat, possibly non-smooth, points in its boundary. We estimate the $L^{p}$ norm of the discrepancy with respect to the translation variable as the dilation parameter goes to infinity. If there is a single flat point with normal in a rational direction we obtain an asymptotic expansion for this norm. Anomalies may appear when two flat points have opposite normals. When all the flat points have normals in generic irrational directions, we obtain a smaller discrepancy. Our proofs depend on careful estimates for the Fourier transform of the characteristic function of the convex body.

math.FA↗