Effective Counting and Spiralling of Lattice Approximates

Nathan Hughes

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 $|q\mathbf{x}-\mathbf{p}|\leq cq^{-\frac{…