Effective Counting and Spiralling of Lattice Approximates

Given $d\ge 2$, we show that the number of approximates $\frac{1}{q}\mathbf{p}\in {\mathbb{Q}}^d$ of $\mathbf{x}\in {\mathbb{R}}^d$ satisfying $\mathrm{\mid }q\mathbf{x}-\mathbf{p}\mathrm{\mid }\le c{q}^{-\frac{1}{d}}$ with denominator $1\le q<T$ 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 $ϵ>0$. Results with the same order are proven for primitive lattice approximates for all $d\ge 1$ 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.