A rational periodicity theorem for division codes (Q630239)

The main subject of this note is normal division code (NDC) of a set, defined as follows: If \(\Phi\subset\mathbb R^m\) is a bounded set and \(d_n^m(\Phi)\) is the greatest lower bound of positive real numbers \(x\) such that \(\Phi\) can be completely covered by \(n\) sets of diameter at most \(x\), then the sequence \(e_n^m(\Phi)=n^{1/m}\cdot d_n^m(\Phi)\) is called the NDC of the set \(\Phi\). For \(m=1\), the NDC of any set has limit \(1\). The author conjectures the existence of a limit for any positive integer \(m\). The notes [\textit{V. P. Filimonov}, Dokl. Math. 79, No. 3, 345--348 (2009); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 426, No. 2, 173--176 (2009; Zbl 1184.28008) and Dokl. Math. 81, No. 2, 278--281 (2010); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 431, No. 5, 593--597 (2010; Zbl 1209.28003)] and this one are attempts to prove this conjecture. Here the following theorem is stated: The rational periodicity theorem: For any bounded set \(\Phi\subset\mathbb R^m\) whose closure is Jordan measurable and has nonzero measure, and any rational number \(q>0\), \(\lim_{n\to\infty}\frac{e^m_{[qn]}(\Phi)}{e_n^m(\Phi)}=1\). The conjecture is important because the existence of a limit for normal division codes is equivalent to the existence of an optimal covering method. After some particular considerations for the case \(m=2\), the author concludes that the properties currently developed for NDC are not enough for proving the proposed conjecture.