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We prove a general result on irregularities of distribution for Borel sets intersected with bounded measurable sets or affine half-spaces.

We consider a planar convex body C and we prove several analogs of Roth's theorem on irregularities of distribution. When the boundary of C is C_2 we prove that for every set of N points in the unit square we have the sharp lower bound N^{1/4} for the L^2 discrepancy. When the boundary is only piecewise C_2 and is not a polygon we have the sharp lower bound N^{1/5}. We also give a whole range of intermediate sharp results between N^{1/5} and N^{1/4}. 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.

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.

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 Lp discrepancy with respect to certain classes of subsets, for example, metric balls.

Trascurare o minimizzare il ruolo della Matematica di Euclide nella scuola secondaria di secondo grado può essere un grave errore: un impoverimento culturale per tutti, e un effettivo handicap" per chi si iscriverà ad un corso di laurea di carattere tecnico-scientifico.

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.

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 is locally flat of finite order. We consider both integer points problems and irregularities of distribution problems. The paper is self-contained. The results on irregularities of distribution are new.

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 Lp 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, for certain values of p, an asymptotic expansion for this norm. Anomalies may appear when two flat points have opposite normals. Our proofs depend on careful estimates for the Fourier transform of the characteristic function of the convex body.

We produce low-discrepancy infinite sequences which can be used to approximate the integral of a smooth periodic function restricted to a smooth convex domain with positive curvature in R^d. The proof depends on simultaneous Diophantine approximation and on appropriate estimates of the decay of the Fourier transform of characteristic functions.

We produce explicit low-discrepancy infinite sequences which can be used to approximate the integral of a smooth periodic function restricted to a convex domain with positive curvature in R2. The proof depends on simultaneous diophantine approximation and a general version of the Erdos-Turán inequality.

Revisiting and extending a recent result of M. Huxley, we estimate the Lp (Td ) and Weak-Lp (Td ) norms of the discrepancy between the volume and the number of integer points in translated domains.

In Geometric Discrepancy we usually test a distribution of N points against a suitable family of sets. If this family consists of dilated, translated and rotated copies of a given d-dimensional convex body D subset of [0, 1)(d), then a result proved by W. Schmidt, J. Beck and H. Montgomery shows that the corresponding L-2 discrepancy cannot be smaller than c(d)N((d-1)/2d). Moreover, this estimate is sharp, thanks to results of D. Kendall, J. Beck and W. Chen. Both lower and upper bounds are consequences of estimates of the decay of parallel to(chi) over cap (D) (rho.)parallel to(L2(Sigma d-1)) for large rho, where (chi) over cap (D) is the Fourier transform (expressed in polar coordinates) of the characteristic function of the convex body D, while Sigma(d-1) is the unit sphere in R-d. In this chapter we provide the Fourier analytic background and we carefully investigate the relation between the L-2 discrepancy and the estimates of parallel to(chi) over cap (D) (rho.)parallel to(L2(Sigma d-1)).

We study the error in quadrature rules on a compact manifold. Our estimates are in the same spirit of the Koksma-Hlawka inequality and they depend on a sort of 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.