Marzo 2021 discrepancy minimal riesz energy points

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theorem_1_1: Marzo and Mas's bound on the spherical cap discrepancy of N-point minimizers of the Riesz s-energy on the d-sphere, of order N^{-2/(d(d-s+1))} for 0 <= s <= d-2 and N^{-2(d-s)/(d(d-s+4))} for d-2 < s < d.

theorem_1_5: Marzo and Mas's two-sided estimate for the Sobolev discrepancy of N-point minimizers of the Riesz s-energy on the d-sphere, between N^{-1/2+s/(2d)} and N^{-1/d} + N^{-1/2+s/(2d)}, sharp for d-2 <= s < d.

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Marzo, Jordi and Mas, Albert, Discrepancy of minimal Riesz energy points. Constr. Approx. 54 (2021), 473--506. DOI 10.1007/s00365-021-09534-5.

Theorem 1.1 bounds the spherical cap discrepancy of an N-point minimizer X_N of the Riesz s-energy on the d-sphere by N^{-2/(d(d-s+1))} for 0 <= s <= d-2 and by N^{-2(d-s)/(d(d-s+4))} for d-2 < s < d, with constants depending only on d and s. This improves Brauchart's bound (display (1.4)) of order N^{-(d-s)/(d(d-s+2))} in the range 0 <= s < 2 for d = 2 (with s = 0 recovering Wolff's unpublished N^{-1/3} bound for logarithmic energy on the 2-sphere) and in the range d - t_0 < s < d for d >= 3, where t_0 = (1+sqrt(17))/2 is about 2.56; in the harmonic case s = d-1 Götz's O(N^{-1/d} log N) bound remains the best. The method follows Wolff: the cap discrepancy is deduced (Proposition 5.2) from Theorem 1.5, an estimate for a Sobolev discrepancy D^eps_{s,d}(X_N) defined (Definition 1.3) as a negative-order Sobolev norm of a smoothed counting measure, expanded in spherical harmonics; the paper calls that estimate sharp for d-2 <= s < d. The paper stresses that all these bounds remain far from Beck's optimal order N^{-(d+1)/(2d)}, up to a logarithmic term, for N-point sets on the d-sphere. The paper does not mention Erdős's problems. The point sets of problem 991, which maximize the product of mutual distances on S^2, are the minimizers of the logarithmic energy, the case d = 2, s = 0 of Theorem 1.1, which bounds their cap discrepancy by a constant times N^{-1/3}.

Source: https://arxiv.org/abs/1907.04814. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1907.04814), every other right reserved. The edition read is arXiv:1907.04814v1 (10 July 2019, 29 pp.); the labels and page numbers on the result pages are that edition's, and were not compared with the journal version.

Bears on. #991: Theorem 1.1 with d = 2, s = 0 applies to the problem's maximizers and gives max_C ||A cap C| - alpha_C n| = O(n^{2/3}), which is o(n) (p. 3).

Results.

• Theorem 1.1 (p. 3): for 0 <= s < d and an N-point minimizer X_N of the Riesz s-energy on S^d, the supremum over spherical caps D of |#(X_N cap D)/N - sigma(D)|, sigma normalized, is at most a constant depending on d and s times chi_{[0,d-2]}(s) N^{-2/(d(d-s+1))} + chi_{(d-2,d)}(s) N^{-2(d-s)/(d(d-s+4))}; Remark 1.2 states the same bound for K-regular sets.
• Theorem 1.5 (p. 5), with Definition 1.3 (pp. 4--5): for such X_N and every small enough eps > 0, the Sobolev discrepancy D^eps_{s,d}(X_N) lies between constant multiples of N^{-1/2+s/(2d)} and N^{-1/d} + N^{-1/2+s/(2d)}.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.