Aug 2026· 4 citations· ⚡ 3 influential· 18 references
Mathematics
Abstract
We prove a nine-distance theorem for Kronecker sequences on flat three-tori. That is, we show that among the first $N$ orbit points, at most nine distinct positive nearest-neighbour distances occur. This proves the conjecture of Haynes and Marklof. An example of Dettmann shows that nine is optimal. More generally, we prove that on a flat $d$-dimensional torus the number of such distances is at most $2^d+1$. The main tool is a new growth theorem for the denominators $q_1<q_2<\cdots$ of best simultaneous approximations in a $d$-dimensional inner-product space, which is of independent interest. We prove that, whenever $q_{n+2^d}$ is defined, either $q_{n+2^d}\ge2q_{n+1}$, or the indices $1,\ldots,2^d$ can be partitioned into disjoint pairs $\{j,k\}$, $j<k$, such that $q_{n+k}=q_n+q_{n+j}$. In particular, $$ q_{n+2^d}\ge \min\{2q_{n+1},q_n+q_{n+2^{d-1}}\}\ge q_n+q_{n+1}. $$
The three gap theorem states that the points of the Kronecker sequence $\alpha,2\alpha,\ldots,N\alpha$, considered modulo one, divide the circle into intervals of at most three distinct lengths. In a two-dimensional nearest-neighbour analogue, Haynes and Marklof proved that the Kronecker sequence modulo an arbitrary un...
The classical three-gap theorem says that a finite Kronecker sequence on the circle has at most three gap lengths. We extend this phenomenon to nearest-neighbour distances on quotients $(V\times U)/\Lambda$, where $V$ is a finite-dimensional real normed space, $U$ is an arbitrary ultrametric abelian group and $\Lambda$...
Let $C_n$ be a cyclic group of order $n$. We prove that if $(n,6)=1$, then every minimal zero-sum sequence of length four over $C_n$ has index one, thereby resolving the length-four index conjecture. After the gcd reduction, the nonunit case follows from the theorem of Shen-Xia-Li, and the remaining unit case is solved...
Hong-Jian Li, Ping-Zhi Yuan, Shijie Yuan et al.· 0 citations
The degree-distortion tradeoff for polynomial approximation of the $d$-dimensional cross-polytope $B_1^d$ is determined, and degree $\Theta(d)$ is necessary and sufficient for constant distortion.
Let $F$ be a field, let $0\le \ell\le k-2$, and suppose that $n\ge k-1$. We determine the minimum degree of a polynomial in $F[x_1,\ldots,x_n]$ that vanishes to order at least $k$ at every nonzero vertex of the Boolean cube and to order exactly $\ell$ at the origin. The answer is \[ n+2k-2-\rho_F(k-\ell), \] where $\rh...
We study the combinatorial (red-blue) discrepancy of finite point sets with respect to hyperplanes and, more generally, bounded-complexity affine algebraic sets. We prove that every $n$-point set in a real Euclidean space admits a red-blue coloring for which every affine algebraic set of dimension at most $D$ and degre...
A. Adıbelli, István Tomon· 1 citation
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