Kourovka Notebook Problem 9.45

Definitions and mathematical statement

Let $n\geq 1$ be an integer and let $a=(a_1,\ldots,a_n)\in{\mathbb{Q}}^n\subseteq{\mathbb{R}}^n$. Define \[ S(a)=\{ka+b:k\in{\mathbb{Z}},\ b\in{\mathbb{Z}}^n\}\subseteq{\mathbb{R}}^n, \qquad \langle x,y\rangle=\sum_{j=1}^n x_jy_j. \] As stated in the source, $S(a)$ is a discrete additive subgroup of ${\mathbb{R}}^n$ of rank $n$. An orthogonal basis of this subgroup means a ${\mathbb{Z}}$-basis $v_1,\ldots,v_n\in S(a)$ such that \[ S(a)=\bigoplus_{i=1}^n{\mathbb{Z}} v_i, \qquad v_i\ne 0\quad(1\leq i\leq n), \qquad \langle v_i,v_j\rangle=0\quad(i\ne j). \] Find necessary and sufficient conditions, expressed in terms of the rational coordinates $a_1,\ldots,a_n$, for such an orthogonal basis to exist.

The supplied registry labels this problem open. Its accompanying discussion reports an explicitly unverified AI claim that conditions have been found; no such conditions are supplied here, and that claim is not treated as an established solution.

Short English statement

Determine exactly which rational vectors $a$ make the lattice ${\mathbb{Z}}^n+{\mathbb{Z}} a$ admit a basis over ${\mathbb{Z}}$ whose vectors are pairwise orthogonal for the standard Euclidean scalar product.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700578$, exact source code catalogue entry, title Kourovka Notebook Problem 9.45.
• Yu. D. Popov and I. V. Protasov, problem 9.45, The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984). The supplied record identifies the later source as the 21st issue edition (October 2026), arXiv:1401.0300v48; its collection description names E. I. Khukhro and V. D. Mazurov as editors.
• Supplied discussion: [Unverified AI claim] that conditions were found, attributed to M. Kinyon and J. Urban, Preprint, 2026: https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka-mathematics.pdf. The supplied status review says this claim is marked with $\#$ in the 21st issue and classifies its evidence as unrefereed_claim.
• The supplied status assessment is dated 2026-10-16 and retains the canonical label open. This is supplied dated curation, not an independently verified literature assessment; no cited source was visited for this draft.