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 unimodular lattice determines at most five distinct nearest-neighbour distances in the Euclidean norm, and that this bound is sharp. Dettmann subsequently constructed examples attaining five distinct distances for every $\ell_p$-norm, $1\leq p\leq\infty$. We prove the corresponding upper bound for every norm on $\mathbb R^2$: for every full-rank lattice $L$, every $\boldsymbol{\alpha}\in\mathbb R^2$, and every $N\in\mathbb N$, the number of distinct nearest-neighbour distances is at most five. For strictly convex norms, the proof extends the lattice-theoretic argument of Haynes and Marklof by replacing the Euclidean angular estimates with a cone lemma based on a proper Brass angular measure. The result for arbitrary norms is then obtained by a strictly convex perturbation and a limiting argument.
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 p...
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 $M_t$ denote the normalized average over the lattice points in the Euclidean ball of radius $t$ in $\mathbb{Z}^d$. We prove that the full maximal operator $f\mapsto\sup_{t\geq0}\lvert M_t f\rvert$ is bounded on $\ell^p(\mathbb{Z}^d)$, for every $1<p\leq\infty$, with a constant independent of the dimension. In parti...
A finite set of unit vectors is $k$-irreducible if every signed sum of between two and $k$ distinct elements has norm greater than one. Let $\mathcal{R}_k$ be the maximal asymptotic rate of such sets, and let $\kappa(\alpha)$ be the maximal asymptotic rate of spherical codes with pairwise inner products at most $\alpha...
For every finite family of Euclidean balls in $\mathbb{R}^n$, $n\ge2$, and every $0<\delta<1/2$, we select a subfamily whose $(1+\delta)$-dilations cover the original union and whose undilated balls have multiplicity at most $C_n\delta^{-(n-1)/2}$. This proves the covering estimate conjectured by Han and Lu \cite{HL}....
Our main result is a $3\sqrt{2\pi}$ bound for the Koml\'os signing problem: every finite family of real vectors of Euclidean norm at most one admits a signed sum of $\ell_\infty$-norm less than this constant, independently of the dimension and the family size. For any $\kappa\ge0$, if a bounded open convex set supports...
