 They gave upper bounds of O( \epsilon ^{-(d+1)/2}) for the minimum lightness in dimensions d\geq 3, and O( \epsilon ^{-(d-1))/2}) for the minimum sparsity in d-space for all d\geq 1.
<meaning-changed>  They gave upper bounds of O( \epsilon ^{-(d+1)/2}) for the minimum lightness in dimensions d\geq 3, and O( \epsilon ^{-(d-1))/2}) for the minimum sparsity in d-space for all d\geq 1.
 They gave upper bounds of O( \epsilon ^{-(d+1)/2}) for the minimum lightness in dimensions d\geq 3, and O( \varepsilon ^{-(d-1))/2}) for the minimum sparsity in d-space for all d\geq 1.