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Giancarlo Travaglini

Publications and source records attributed to Giancarlo Travaglini.

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 $μ$, 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 $μ$ depends on $M$. In this latter context, we prove that, up to a constant, the lowest possible discrepancy is achieved when the support of $μ$ has diameter $\leq c/M$, and that this upper bound is sharp.

math.NT↗
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↗
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 \[ \int_{0}^{1}\int_{\mathbb{T}^{2}}\left\vert \mathrm{card}\left( \mathcal{P}_{N}\mathcal{\cap}\left( λC+t\right) \right) -λ^{2}N\left\vert C\right\vert \right\vert ^{2}~dtdλ\geqslant cN^{1/2}\;. \] When $\partial C$ is only piecewise $\mathcal{C}^{2}$ and is not a polygon we prove the sharp bound% \[ \int_{0}^{1}\int_{\mathbb{T}^{2}}\left\vert \mathrm{card}\left( \mathcal{P}_{N}\mathcal{\cap}\left( λC+t\right) \right) -λ^{2}N\left\vert C\right\vert \right\vert ^{2}~dtdλ\geqslant cN^{2/5}. \] 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↗
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↗
arXiv2017

Discrepancy and numerical integration on metric measure spaces

We study here the error of numerical integration on metric measure spaces adapted to a decomposition of the space into disjoint subsets. We consider both the error for a single given function, and the worst case error for all functions in a given class of potentials. The main tools are the classical Marcinkiewicz-Zygmund inequality and ad hoc definitions of function spaces on metric measure spaces. The same techniques are used to prove the existence of point distributions in metric measure spaces with small $L^p$ discrepancy with respect to certain classes of subsets, for example metric balls.

math.AP↗
arXiv2015

A characterization theorem for the $L^{2}$-discrepancy of integer points in dilated polygons

Let $C$ be a convex $d$-dimensional body. If $ρ$ is a large positive number, then the dilated body $ρC$ contains $ρ^{d}\left\vert C\right\vert +\mathcal{O}\left( ρ^{d-1}\right) $ integer points, where $\left\vert C\right\vert $ denotes the volume of $C$. The above error estimate $\mathcal{O}\left( ρ^{d-1}\right) $ can be improved in several cases. We are interested in the $L^{2}$-discrepancy $D_{C}(ρ)$ of a copy of $ρC$ thrown at random in $\mathbb{R}^{d}$. More precisely, we consider \[ D_{C}(ρ):=\left\{ \int_{\mathbb{T}^{d}}\int_{SO(d)}\left\vert \textrm{card}\left( \left( ρσ(C)+t\right) \cap\mathbb{Z}^d\right) - ρ^{d}\left\vert C\right\vert \right\vert ^{2}dσdt\right\} ^{1/2}\ , \] where $\mathbb{T}^{d}=$ $\mathbb{R}^{d}/\mathbb{Z}^{d}$ is the $d$-dimensional flat torus and $SO\left( d\right) $ is the special orthogonal group of real orthogonal matrices of determinant $1$. An argument of D. Kendall shows that $D_{C}(ρ)\leq c\ ρ^{(d-1)/2}$. If $C$ also satisfies the reverse inequality $\ D_{C}(ρ)\geq c_{1} \ ρ^{(d-1)/2}$, we say that $C$ is $L^{2}$\emph{-regular}. L. Parnovski and A. Sobolev proved that, if $d>1$, a $d$-dimensional unit ball is $L^{2}% $-regular if and only if $d\not \equiv 1\ (\operatorname{mod}4)$. In this paper we characterize the $L^{2}$-regular convex polygons. More precisely we prove that a convex polygon is not $L^{2}$-regular if and only if it can be inscribed in a circle and it is symmetric about the centre.

math.NT↗
arXiv2010

Quadrature rules and distribution of points on manifolds

We study the error in quadrature rules on a compact manifold. As in the Koksma-Hlawka inequality, we consider a discrepancy of the sampling points and a generalized variation of the function. In particular, we give sharp quantitative estimates for quadrature rules of functions in Sobolev classes.

math.NT↗
arXiv2010

An L^1 estimate for half-space discrepancy

For every unit vector $σ\inΣ_{d-1}$ and every $r\ge0$, let % % \begin{displaymath} P_{σ,r}=[-1,1]^d\cap\{t\in\Rr^d:t\cdotσ\le r\} \end{displaymath} % % denote the intersection of the cube $[-1,1]^d$ with a half-space containing the origin $0\in\Rr^d$. We prove that if $N$ is the $d$-th power of an odd integer, then there exists a distribution $\PPP$ of $N$ points in $[-1,1]^d$ such that % % \begin{displaymath} \sup_{r\ge0} \int_{Σ_{d-1}}\vert\card(\PPP\cap P_{σ,r})-N2^{-d} \vert P_{σ,r}\vert\vert\,\ddσ\le c_d(\log N)^d, \end{displaymath} % % generalizing an earlier result of Beck and the first author.

math.NT↗
arXiv2010

Trigonometric approximation and a general form of the Erdős Turán inequality

There exists a positive function $ψ(t)${on}$t\geq0${, with fast decay at infinity, such that for every measurable set}$Ω${in the Euclidean space and}$R>0${, there exist entire functions}$A(x) ${and}$B(x) ${of exponential type}$R${, satisfying\}$A(x)\leq χ_Ω(x)\leq B(x)${and}$| B(x)-A(x)| \leqslantψ(R\operatorname*{dist}(x,\partialΩ)) $. This leads to Erdős Turán estimates for discrepancy of point set distributions in the multi dimensional torus. Analogous results hold for approximations by eigenfunctions of differential operators and discrepancy on compact manifolds.

math.NT↗
arXiv2010

Convolution operators defined by singular measures on the motion group

This paper contains an $L^{p}$ improving result for convolution operators defined by singular measures associated to hypersurfaces on the motion group. This needs only mild geometric properties of the surfaces, and it extends earlier results on Radon type transforms on $\mathbb{R}^{n}$. The proof relies on the harmonic analysis on the motion group.

math.FA↗