Nearly linear lattice coverings of arbitrary convex bodies

The problem asks for the smallest universal order of lattice-covering density for every convex body in $\mathbb{R}^n$. Heng Li and Xizhi Liu’s paper determines the optimal polynomial growth exponent, but not the sharp order including subpolynomial factors.

Known results

• Coxeter, Few, and Rogers: Euclidean balls have density at least $(e^{-3/2}+o(1))n$.
• Ordentlich, Regev, and Weiss, 2021: every convex body has lattice-covering density $O(n^2)$.
• Schymura, Wang, and Xue conjectured a universal bound $Cn(\log n)^{1+\log_2 e}$.
• For Euclidean balls, the previous bound was $O(n(\log n)^{1.85837\ldots})$.

July 2026 near-linear bound

Li and Liu prove, for every sufficiently large $n$ and every $n$-dimensional convex body $K$,

$$\theta_L(K)\leq Cn\log n\,(\log\log n)^{10/3+o(1)}.$$

Combined with the ball lower bound, this gives $\Theta_n^{\rm conv}=n^{1+o(1)}$ and confirms the conjecture at polynomial scale. The exact smallest order remains open.

Current status (as of July 2026): The optimal polynomial exponent is settled at $1$, while the sharp universal order and logarithmic factors remain open.