Protasov 2024 optimal partitions flat torus into parts

../

theorem_1: Bounds the least maximal part diameter d_m(T^2) of an m-part partition of the flat torus above by sqrt(1/4 + 1/m^2) and below by 2/sqrt(pi m) for m >= 6 and by 1/k when m = k^2 + k - 1.

theorem_2: Determines d_1(T^2) = d_2(T^2) = sqrt(2)/2 and d_3(T^2) = sqrt(13)/6 for partitions of the flat torus into parts of least maximal diameter.

theorem_3: Gives upper and lower bounds on d_m(T^2) for 4 <= m <= 7, the lower ones from SAT-certified non-colorability of torus grid graphs and the upper ones from explicit partitions, with a misprinted closed form for the m = 7 upper bound.

---

D. S. Protasov, A. D. Tolmachev, V. A. Voronov, Optimal partitions of the flat torus into parts of smaller diameter, arXiv preprint (2024), arXiv:2402.03997v1 [math.MG], dated 6 February 2024, 18 pages. The copy read for this card is that arXiv v1 PDF; its labels and page numbers are the ones cited here. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2402.03997), every other right reserved.

The paper studies $d_m(T^2)$, the infimum of the $x$ for which the flat torus $T^2=\mathbb R^2/\mathbb Z^2$, with the metric induced from the plane, is the union of $m$ parts each of diameter at most $x$, a Borsuk-type quantity (pp. 1--2). Theorem 1 (p. 2) gives elementary bounds: $d_m(T^2)\le\sqrt{1/4+1/m^2}$ for all $m$, $d_m(T^2)\ge2/\sqrt{\pi m}$ for $m\ge6$, and $d_{k^2+k-1}(T^2)\ge1/k$. Theorem 2 (p. 3) settles the small cases exactly: $d_1(T^2)=d_2(T^2)=\sqrt2/2$ and $d_3(T^2)=\sqrt{13}/6$, the last by a covering argument on vertical and horizontal lines (Propositions 1--6, pp. 6--9) that uses Raikov's inequality for sums of closed subsets of the circle (Theorem 4, p. 6, cited, not proved). The introduction (p. 2) notes that $T^n$ splits into three layers of thickness $\frac13$ of diameter below $\operatorname{diam}T^n$, so the Borsuk number of $T^n$ is $3$ for all $n\ge1$.

Theorem 3 (p. 3) bounds $d_m(T^2)$ for $4\le m\le7$, closely for $m=4,5$ (for example $\sqrt{12401}/200\le d_4(T^2)\le\sqrt5/4$) and loosely for $m=6,7$. Its lower bounds rest on SAT-solver reports that grid graphs on the torus have no proper $m$-coloring (Propositions 7 and 8, Table 2, pp. 10--11), and its upper bounds for $m=5,6,7$ on partitions checked by a validation script in the authors' repository (p. 9). As printed, the $m=7$ upper bound reads $(5-\sqrt7)/3=0.511452\ldots$; the closed form equals $0.7847\ldots$, and the decimal, which Table 1 repeats, is $\sqrt{5-\sqrt7}/3$. Table 1 (p. 4) lists numerical bounds for $1\le m\le25$, the upper ones for $m\ge8$ from optimized periodic hexagonal tilings (Section 3.3.3, Table 3, pp. 11--14) and from gradient-based optimization of polygonal partitions (Section 3.3.4, pp. 14--15); its $m=4$ lower bound is printed as $0.556707$, against $0.556799\ldots$ in Theorem 3. For $m\ge4$ the paper does not determine $d_m(T^2)$ (p. 3), and its Question 1 (p. 15) asks whether the $m=4,5,6$ estimates of Theorem 3 are exact.

Source: https://arxiv.org/abs/2402.03997.

Results. Labels and pages are the print's. Each statement was read clause by clause against the print (claims checked); no proof was checked step by step, and the computations behind Theorem 3 were not rerun.

• Theorem 1 (p. 2; proof pp. 4--5): the strip upper bound $\sqrt{1/4+1/m^2}$, the area lower bound $2/\sqrt{\pi m}$ for $m\ge6$, and $d_{k^2+k-1}(T^2)\ge1/k$.
• Theorem 2 (p. 3; proof pp. 5--9): $d_1(T^2)=d_2(T^2)=\sqrt2/2$ and $d_3(T^2)=\sqrt{13}/6$.
• Theorem 3 (p. 3; verification pp. 9--11): computer-assisted upper and lower bounds on $d_m(T^2)$ for $4\le m\le7$, with the $m=7$ misprint recorded.

Bears on.

• Problem 508: no result of the paper concerns the chromatic number of the plane. The connection is of method only: the lower bounds of Theorem 3 rest on SAT-solver reports that grid graphs on the torus, whose edges join points at distance at least $\tau$, have no proper coloring with $m$ colors, and the paper cites SAT colorings of unit-distance strips in the setting of the Hadwiger--Nelson problem as a similar use of the method (p. 11).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.