Optimal bounds for design curves on the torus
Abstract
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 labeleq:abs L≥ (((t+1)2d−1)/(t(t+2)))1/2. Equality is attained by the family of Korobov curves described in \cite{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.
Related
- Optimal asymptotic bounds for design curves on manifolds
- Triple intersections for algebraic curves on the real torus
- Improved exponent bounds for Erdős--Selfridge curves
- Explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields
- Explicit bounds for the number of solutions to Roth-type inequalities over adelic curves
BibTeX
@misc{ehler2026optimal,
title = {{Optimal bounds for design curves on the torus}},
author = {Martin Ehler},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03362},
eprint = {2611.03362},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03362},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal bounds for design curves on the torus
AU - Martin Ehler
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03362
UR - https://arxiv.org/abs/2611.03362
AB - A closed rectifiable curve on the torus 𝕋^d=ℝ^d/ℤ^d is a t-design curve if normalized arclength integration is exact for trigonometric polynomials e^2πi ⟨ m,x⟩ with frequencies m∈ℤ^d satisfying |m|_∞≤ t. For the arclength L of such curves, we establish the sharp lower bound labeleq:abs L≥ (((t+1)^2d−1)/(t(t+2)))^1/2. Equality is attained by the family of Korobov curves described in \cite{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 t^2-design curves in dimension two. For d≥3, we establish lower bounds of optimal order for the length of hyperbolic cross t^d-design curves. Finally, we obtain lower bounds of optimal order for the length of t-design curves associated with sparse frequency sets.
ER -