Effective Counting and Spiralling of Lattice Approximates

TL;DR

This paper establishes precise asymptotic counts for lattice approximates in multiple dimensions, improving previous results especially for primitive approximates and linear forms, with detailed error bounds.

Contribution

It provides new asymptotic formulas and error estimates for counting lattice approximates, extending and refining Schmidt's earlier work, including the primitive case and affine lattices.

Findings

Asymptotic count of lattice approximates with explicit error bounds

Results valid for all dimensions and primitive approximates

Improves upon Schmidt's results, especially for d=1

Abstract

Given $d\geq 2$, we show that the number of approximates $\frac{1}{q}\mathbf{p}\in \mathbb{Q}^d$ of $\mathbf{x}\in\mathbb{R}^d$ satisfying $|q\mathbf{x}-\mathbf{p}|\leq cq^{-\frac{1}{d}}$ with denominator $1\leq q < T$ decays to the asymptotic term $c\text{vol}_d(B_d(0,1))\log T$ with error of order $\left(\log T\right)^{-\frac{1}{2}}\left(\log \log T\right)^\frac{3}{2}\left(\log\log\log T\right)^{\frac{1}{2}+\epsilon}$ for almost all $\mathbf{x}\in\mathbb{R}^d$ and for any $\epsilon >0$. Results with the same order are proven for primitive lattice approximates for all $d\geq 1$ and also for the case of linear forms and affine lattices. These results, especially in the primitive case for $d=1$, are an improvement to the results of Schmidt.