We study point configurations on the torus 𝕋^d that minimize interaction energies with tensor product structure. Such interactions arise naturally in the context of discrepancy theory and quasi-Monte Carlo integration. Permutation sets on 𝕋^2 and Latin hypercube sets in higher dimensions (i.e. sets whose projections onto coordinate axes are equispaced points) are natural candidates to be energy minimizers. We show that such point configurations that have only one distance in the vector sense minimize the energy for a wide range of potentials, i.e. such sets satisfy a tensor product version of universal optimality. This specifically applies to three- and five-point Fibonacci lattices. We also characterize all lattices with this property and exhibit some non-lattice sets of this type. In addition, we obtain several further structural results about global and local minimizers of tensor product energies.

We investigate the asymptotic behavior of greedy s-Riesz and Green energy sequences {x_n}_n=1^∞ on the unit sphere 𝕊^d⊂ℝ^d+1, where each point x_n is defined as the minimizer of the discrete potential generated by the preceding points x_1, x_2, ..., x_n-1. We show that the greedy sequence attains optimal growth behavior for the second-order term of the Green and Riesz s-energies when d-2 ≤ s < d. The main idea is to establish the bounds on polarization using well-separation properties of the greedy configurations.

We consider random lines in ℝ^3 (random with respect to the kinematic measure) and how they intersect 𝕊^2 . It is known that the entry point and the exit point behave like independent uniformly distributed random variables. We give a new proof using bilinear integral geometry and use this approach to show that this property is extremely rare: if K ⊂ℝ^n is a bounded, convex domain with smooth boundary with this property (i.e., the intersection points with a random line are independent), then n=3 and K is a ball.

We study measures and point configurations optimizing energies based on multivariate potentials. The emphasis is put on potentials defined by geometric characteristics of sets of points, which serve as multi-input generalizations of the well-known Riesz potentials for pairwise interaction. One of such potentials is volume squared of the simplex with vertices at the $k \ge 3$ given points: we show that the arising energy is maximized by balanced isotropic measures, in contrast to the classical two-input energy. These results are used to obtain interesting geometric optimality properties of the regular simplex. As the main machinery, we adapt the semidefinite programming method to this context and establish relevant versions of the $k$-point bounds.

We start by providing a very simple and elementary new proof of the classical bound due to J. Beck which states that the spherical cap 𝕃_2-discrepancy of any N points on the unit sphere 𝕊^d in ℝ^d+1, d≥2, is at least of the order N^-1/2-1/2d. The argument used in this proof leads us to many further new results: estimates of the discrepancy in terms of various geometric quantities, an easy proof of point-independent upper estimates for the sum of positive powers of Euclidean distances between points on the sphere, lower bounds for the discrepancy of rectifiable curves and sets of arbitrary Hausdorff dimension. Moreover, refinements of the proof also allow us to obtain explicit values of the constants in the lower discrepancy bound on 𝕊^d. The value of the obtained asymptotic constant falls within 3% of the conjectured optimal constant on 𝕊^2 (and within up to 7% on 𝕊^4, 𝕊^8, 𝕊^24).

It is well understood that if one is given a set X ⊂ [0,1] of n independent uniformly distributed random variables, then sup _0 ≤ x ≤ 1| # X ∩ [0,x]/# X - x | ≲√(logn)/√(n) with high probability. We show that one can improve the error term by removing a few of the points. For any m ≤ 0.001n there exists a subset Y ⊂ X obtained by deleting at most m points, so that the error term drops from ∼√(logn)/√(n) to log(n)/m with high probability. When m=cn for a small 0 ≤ c ≤ 0.001 , this achieves the essentially optimal asymptotic order of discrepancy log (n)/n . The proof is constructive and works in an online setting (where one is given the points sequentially, one at a time, and has to decide whether to keep or discard it). A change of variables shows the same result for any random variables on the real line with absolutely continuous density.

We study probability measures that minimize the Riesz energy with respect to the geodesic distance ϑ (x,y) on projective spaces 𝔽ℙ^d (such energies arise from the 1959 conjecture of Fejes Tóth about sums of non-obtuse angles), i.e. the integral 1/s∫_𝔽ℙ^d∫_𝔽ℙ^d( ϑ (x,y) )^-s dμ(x) dμ (y) for s<d and find ranges of the parameter s for which the energy is minimized by the uniform measure σ on 𝔽ℙ^d. To this end, we use various methods of harmonic analysis, such as Cesàro averages of Jacobi expansions and A_1 inequalities, and establish a rather general theorem guaranteeing that certain energies with singular kernels are minimized by σ. In addition, we obtain further results and present numerical evidence, which uncover a peculiar effect that minimizers this energy undergo numerous phase transitions, in sharp contrast with many analogous known examples (even the seemingly similar geodesic Riesz energy on the sphere), which usually have only one transition (between uniform and discrete minimizers).

This paper is devoted to spherical measures and point configurations optimizing three-point energies. Our main goal is to extend the classic optimization problems based on pairs of distances between points to the context of three-point potentials. In particular, we study three-point analogues of the sphere packing problem and the optimization problem for p p -frame energies based on three points. It turns out that both problems are inherently connected to the problem of nearly orthogonal sets by Erdős. As the outcome, we provide a new solution of the Erdős problem from the three-point packing perspective. We also show that the orthogonal basis uniquely minimizes the p p -frame three-point energy when 0 > p > 1 0>p>1 in all dimensions. The arguments make use of multivariate polynomials employed in semidefinite programming and based on the classical Gegenbauer polynomials. For p = 1 p=1 , we completely solve the analogous problem on the circle. As for higher dimensions, we show that the Hausdorff dimension of minimizers is not greater than d − 2 d-2 for measures on S d − 1 \mathbb {S}^{d-1} .

A celebrated result of Beck shows that for any set of N points on S-d there always exists a spherical cap B C S-d such that number of points in the cap deviates from the expected value sigma ( B ) N by at least N (1 /2 -1 /2 d) , where sigma is the normalized surface measure. We refine the result and show that, when d (sic)1 (mod 4), there exists a (small and very specific) set of real numbers such that for every r > 0 from the set one is always guaranteed to find a spherical cap C-r with the given radius r for which the result holds. The main new ingredient is a generalization of the notion of badly approximable numbers to the setting of Gegenbauer polynomials: these are fixed numbers x is an element of ( - 1 , 1) such that the sequence of Gegenbauer polynomials ( C (lambda)(n) ( x )) (infinity) (n =1) avoids being close to 0 in a precise quantitative sense. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We show that the lower bound for the optimal directional discrepancy with respect to the class of rectangles in ℝ^2 rotated in a restricted interval of directions [-θ ,θ ] with θ < π/4 is of the order at least N^1/5 with a constant depending on θ .

We investigate the behavior of a greedy sequence on the sphere $\mathbb{S}^d$ defined so that at each step the point that minimizes the Riesz $s$-energy is added to the existing set of points. We show that for $0

We provide a detailed analysis of results from a large-scale computational exploration of real and complex (weighted) point configurations that minimize p-frame energies, uncovering phase transition behavior exhibited by the minimizers. We utilize numerical linear programming methodologies to offer complementary lower bounds that support our experimentally obtained upper bounds on minimal energy values. Furthermore, we present the development of an exceptionally symmetric weighted design consisting of 85 points, which outperforms the current best known lower bounds for a minimal-sized weighted design in the realm of five-dimensional complex projective space. In conclusion, based on our thorough observations and in-depth analysis, we conjecture that the support of this novel weighted design is universally optimal.

In this paper we elaborate on the interplay between energy optimization, positive definiteness, and discrepancy. In particular, assuming the existence of a K-invariant measure μ with full support, we show that conditional positive definiteness of a kernel K is equivalent to a long list of other properties: including, among others, convexity of the energy functional, inequalities for mixed energies, and the fact that μ minimizes the energy integral in various senses. In addition, we prove a very general form of the Stolarsky Invariance Principle on compact spaces, which connects energy minimization and discrepancy and extends several previously known versions.

We provide new answers about the placement of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the $p$-frame energies, i.e. energies with the kernel given by the absolute value of the inner product raised to a positive power $p$. Application of linear programming methods in the setting of projective spaces allows for describing the minimizing measures in full in several cases: we show optimality of tight designs and of the $600$-cell for several ranges of $p$ in different dimensions. Our methods apply to a much broader class of potential functions, those which are absolutely monotonic up to a particular order as functions of the cosine of the geodesic distance. In addition, a preliminary numerical study is presented which suggests optimality of several other highly symmetric configurations and weighted designs in low dimensions. In one case we improve the best known lower bounds on a minimal sized weighted design in $\mathbb{CP}^4$. All these results point to the discreteness of minimizing measures for the $p$-frame energy with $p$ not an even integer.

In the present paper we develop the theory of minimization for energies with multivariate kernels, i.e. energies, in which pairwise interactions are replaced by interactions between triples or, more generally, n-tuples of particles. Such objects, which arise naturally in various fields, present subtle differences and complications when compared to the classical two-input case. We introduce appropriate analogues of conditionally positive definite kernels, establish a series of relevant results in potential theory, explore rotationally invariant energies on the sphere, and present a variety of interesting examples, in particular, some optimization problems in probabilistic geometry which are related to multivariate versions of the Riesz energies.

In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In particular, we prove that the support of any minimizer of the $p$-frame energy has empty interior whenever $p$ is not an even integer. A similar effect is also demonstrated for energies with analytic potentials which are not positive definite. In addition, we establish the existence of discrete minimizers for a large class of energies, which includes energies with polynomial potentials.

This paper is concerned with approximation properties of polynomially enriched wavelet systems, so-called quarklet frames. We show that certain model singularities that arise in elliptic boundary value problems on polygonal domains can be approximated from the span of such quarklet systems at inverse-exponential rates. In order to realize these, we combine spatial refinement in the vicinity of the singularities with suitable growth of the polynomial degrees in regions where the solution is smooth, similar to adaptive hp-finite element approximation.

We survey some of the recent developments in uniform distribution and discrepancy theory, which include, in particular, the fact that Poissonian pair correlation implies uniform distribution, the progress on Tusnady's problem, Levin's lower bounds for the discrepancy of most low-discrepancy sequences, a link between the small ball inequality and digital nets, various heuristic arguments supporting two conflicting conjectures on the growth of the star-discrepancy of d-dimensional points set, and different versions of the Stolarsky principle for the discrepancy on the sphere. We discuss known results and pose some open problems and conjectures.