Kolountzakis–Papageorgiou Question 1 on common tiling functions

The question asks whether some pairwise trivially intersecting lattice families admit common tiling functions whose support grows sublinearly in the number of lattices. It was recorded in the 2021 paper Functions tiling with several lattices.

Known results

• Every common tiling function satisfies the lower bound $\operatorname{diam}\operatorname{supp} f\ge C_dN^{1/d}$.
• For round lattices, convolution gives an $O(N)$ upper bound.
• Special families can force the linear lower bound $\operatorname{diam}\operatorname{supp} f\ge C_dN$.
• Families with algebraic relations admit smaller common tiles, but they do not satisfy $\Lambda_i\cap\Lambda_j=\{0\}$ for $i\ne j$.

August 25, 2026 construction claim

A report dated August 25, 2026 describes an unrefereed preprint, Common tiling functions with small support, claiming pairwise trivially intersecting lattice families with nonnegative common tiling functions of support diameter $O(N^{1/d})$ for all sufficiently large $N$. This matches the lower-bound scale for selected families, not the full admissible-family question, and remains unverified.

Current status (as of August 2026): the general question remains open; the selected-family $O(N^{1/d})$ construction is claimed but unverified.