New asymptotic estimates for spherical designs
arXiv:0811.01687 citationsh-index: 14
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This provides tighter theoretical limits for spherical designs, which are important in approximation theory, numerical integration, and coding theory.
The authors prove new asymptotic upper bounds for the minimal number of points in spherical t-designs on the unit sphere, improving known exponents for dimensions 3 through 10 and providing a general bound for higher dimensions.
Let N(n, t) be the minimal number of points in a spherical t-design on the unit sphere S^n in R^{n+1}. For each n >= 3, we prove a new asymptotic upper bound N(n, t) <= C(n)t^{a_n}, where C(n) is a constant depending only on n, a_3 <= 4, a_4 <= 7, a_5 <= 9, a_6 <= 11, a_7 <= 12, a_8 <= 16, a_9 <= 19, a_10 <= 22, and a_n < n/2*log_2(2n), n > 10.