Single radius spherical cap discrepancy on compact two-point homogeneous spaces
arxiv(2024)
Abstract
In this note we study estimates from below of the single radius spherical
discrepancy in the setting of compact two-point homogeneous spaces. Namely,
given a d-dimensional manifold ℳ endowed with a distance ρ so
that (ℳ, ρ) is a two-point homogeneous space and with the
Riemannian measure μ, we provide conditions on r such that if D_r
denotes the discrepancy of the ball of radius r, then, for an absolute
constant C>0 and for every set of points {x_j}_j=1^N, one has
∫_ℳ |D_r(x)|^2 dμ(x)⩾ C N^-1-1/d. The
conditions on r that we have depend on the dimension d of the manifold and
cannot be achieved when d ≡ 1 ( mod4). Nonetheless, we
prove a weaker estimate for such dimensions as well.
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