Sharper bounds for approximate L2-designs

Improve either the upper or lower bounds on the number of points required by approximate spherical L^2-designs.

Background

The paper introduces L^2-approximate spherical designs and proves separate lower and upper bounds for their sizes. Unlike the tensor-approximate setting, these bounds do not match, leaving a quantitative gap in the dependence on the dimension and approximation parameter.

The authors explicitly pose improving either side of this gap as an unresolved problem.