On Diophantine exponents of lattices

TL;DR

The paper analyzes the spectrum of ordinary Diophantine exponents for $d$-dimensional lattices by defining $\psi_\Lambda(t)$ and $\omega(\Lambda)$ via $\Pi(\mathbf{x})$, establishing trivial lower bounds and known results, and proving that for all $d\ge3$ the spectrum ${\bf S}_d$ is the full interval $[0,\infty]$. The core method reduces the problem to a two-dimensional subspace and uses a metric-approximation framework, embedding a 2D lattice $\Gamma_{\alpha,\beta}$ into a $d$-dimensional lattice and applying a two-stage Borel–Cantelli analysis to obtain both generic lower bounds and precise exceptional asymptotics around Diophantine convergents. The main results are Theorem 1, asserting ${\bf S}_d=[0,\infty]$ for $d\ge3$, and Theorem 2, a general existence principle that yields sharp (in a metric sense) control of $\Pi^d(\mathbf{x})$ in terms of a decaying weight $w$, with connections to exact-order results and recent precise approximation theorems (e.g., Baker–Ward) and Skriganov-type bounds. Altogether, the paper blends geometry of numbers with metric Diophantine approximation to resolve the spectrum issue in higher dimensions and outlines pathways to sharper quantitative statements.

Abstract

We describe the spectrum of ordinary Diophantine exponents for $d$-dimensional lattices. The result reduces the problem to two-dimensional case and uses argument of metric theory.