The classical Stolarsky invariance principle connects the spherical cap L 2 discrepancy of a finite point set on the sphere to the pairwise sum of Euclidean distances between the points. In this paper we further explore and extend this phenomenon. In addition to a new elementary proof of this fact, we establish several new analogs, which relate various notions of discrepancy to different discrete energies. In particular, we find that the hemisphere discrepancy is related to the sum of geodesic distances. We also extend these results to arbitrary measures on the sphere and arbitrary notions of discrepancy and apply them to problems of energy optimization and combinatorial geometry and find that, surprisingly, the geodesic distance energy behaves differently than its Euclidean counterpart.

For a finite abelian group G and positive integers m and h, we let ρ(G, m, h) = min{|hA| : A ⊆ G, |A| = m} and ρ±(G, m, h) = min{|h±A| : A ⊆ G, |A| = m},where hA and h±A denote the h-fold sumset and the h-fold signed sumset of A, respectively. The study of ρ(G, m, h) has a 200-year-old history and is now known for all G, m, and h. Here we prove that ρ±(G, m, h) equals ρ(G, m, h) when G is cyclic, and establish an upper bound for ρ±(G, m, h) that we believe gives the exact value for all G, m, and h.Theorem 1 (Cauchy-Davenport Theorem) If A and B are nonempty subsets of the group Z p of prime order p, then |A + B| ≥ min{p, |A| + |B| − 1}.It can easily be seen that the bound is tight for all values of |A| and |B|, and thusAfter various partial results, the general case was finally solved in 2006 by Plagne [15] (see also [14], [9], and [10]). To state the result, we introduce the functionwhere n, m, and h are positive integers, D(n) is the set of positive divisors of n, and f d (m, h) = (h ⌈m/h⌉ − h + 1) · d. (Here u(n, m, h) is a relative of the Hopf-Stiefel function used also in topology and bilinear algebra; see, for example, [8], [11], [14], and [16].) Theorem 2 (Plagne; cf. [15]) Let n, m, and h be positive integers with m ≤ n. For any abelian group G of order n we have ρ(G, m, h) = u(n, m, h).Turning now to ρ ± (G, m, h), we start by observing that ρ ± (G, m, 0) = 1 and ρ ± (G, m, 1) = m for all G and m. To see the latter equality, it suffices to verify that one can always find a symmetric subset of size m in G, that is, an m-subset A of G for which A = −A. Therefore, from now on, we assume that h ≥ 2.We must admit that our study of ρ ± (G, m, h) resulted in quite a few surprises. For a start, we noticed that, in spite of the fact that h ± A is usually much larger than hA is, the equality ρ ± (G, m, h) = ρ(G, m, h) holds quite often; it is an easy exercise to verify that, among groups of order 24 or less, equality holds with only one exception: ρ ± (Z 2 3 , 4, 2) = 8 while ρ(Z 2 3 , 4, 2) = 7. In fact, we can prove that ρ ± (G, m, h) agrees with ρ(G, m, h) for all cyclic groups G and all m and h (see Theorem 4 below).However, in contrast to ρ(G, m, h), the value of ρ ± (G, m, h) depends on the structure of G rather than just the order n of G. Suppose that G is of type (n 1 , . . . , n r ), that is,

For a finite abelian group G and positive integers m and h, we letwhere hA and h±A denote the h-fold sumset and the h-fold signed sumset of A, respectively. The study of ρ(G, m, h) has a 200-year-old history and is now known for all G, m, and h. In previous work we provided an upper bound for ρ±(G, m, h) that we believe is exact, and proved that ρ±(G, m, h) agrees with ρ(G, m, h) when G is cyclic. Here we study ρ±(G, m, h) for elementary abelian groups G; in particular, we determine all values of m for which ρ±(Z 2 p , m, 2) equals ρ(Z 2 p , m, 2) for a given prime p.

In 1959 Fejes Tóth posed a conjecture that the sum of pairwise non-obtuse angles between N unit vectors in S d is maximized by periodically repeated elements of the standard orthonormal basis. We obtain new improved upper bounds for this sum, as well as for the corresponding energy integral. We also provide several new approaches to the only settled case of the conjecture: d = 1.arXiv:1801.07837v1 [math.MG]

In this paper we study Riesz, Green and logarithmic energy on two-point homogeneous spaces. More precisely we consider the real, the complex, the quaternionic and the Cayley projective spaces. For each of these spaces we provide upper estimates for the mentioned energies using determinantal point processes. Moreover, we determine lower bounds for these energies of the same order of magnitude.