An L¹ estimate for half-space discrepancy

الأصل · EN

For every unit vector σ∈Σd₋₁ and every r≥0, let % % displaymath Pσ,ᵣ=[-1,1]ᵈ∩{t∈ᵈ:t·σ≤ r} displaymath % % denote the intersection of the cube [-1,1]ᵈ with a half-space containing the origin 0∈ᵈ. We prove that if N is the d-th power of an odd integer, then there exists a distribution of N points in [-1,1]ᵈ such that % % displaymath ᵣ≥₀ ∫Σd₋₁(∩ Pσ,ᵣ)-N2⁻ᵈ Pσ,ᵣσ≤ cd(N)ᵈ, displaymath % % generalizing an earlier result of Beck and the first author.