Optimal bounds for design curves on the torus

A closed rectifiable curve on the torus Tᵈ= Rᵈ/ Zᵈ is a t-design curve if normalized arclength integration is exact for trigonometric polynomials e2πi m,x with frequencies m∈ Zᵈ satisfying \|m\|_∞≤ t. For the arclength L of such curves, we establish the sharp lower bound {equation}{eq:abs} L≥ ({(t+1)²ᵈ-1}{t(t+2)})1/2. {equation} Equality is attained by the family of Korobov curves described in {Kunis,Lindblad}. For even t, we also derive an explicit lower bound on the length of total-degree t-design curves, defined by the frequency condition \|m\|₁≤ t. This bound is sharp for $t=2$ in every dimension and for every even t in dimension two. We determine the minimum length of hyperbolic cross t²-design curves in dimension two. For d≥3, we establish lower bounds of optimal order for the length of hyperbolic cross tᵈ-design curves. Finally, we obtain lower bounds of optimal order for the length of t-design curves associated with sparse frequency sets.