Planar lattices and equilateral odd-gons

Abstract

For a planar integral lattice $L$, let $\hat{I}½(L)$ denote the square-free part of the integer $D(L{)}^2$, where $D(L)$ stands for the area of a fundamental parallelogram of $L$. For each odd integer $n$ with $3\le n<29$, a planar lattice $L$ contains an equilateral $n$-gon if and only if $L$ is similar to an integral lattice $L^{\prime }$ such that $\hat{I}½(L^{\prime })\equiv 3(\mathrm{mod}4)$ and the largest prime factor $p$ of $\hat{I}½(L^{\prime })$ satisfies $p\le n$. Moreover, such $L$ contains a convex equilateral $n$-gon, which answers a problem posed by Maehara.

7 pages; v2: updated to incorporate the corrections published in Yokohama Math. J. 71, 61-62 (2025). Main results remain unchanged