= Geometry of numbers
Geometry of numbers uses the geometry of <lattices> in real vector spaces to study arithmetic questions. A full-rank lattice $\Lambda=A\mathbb Z^n\subset\mathbb R^n$ has covolume $|\det A|$, the volume of a fundamental parallelotope. Geometric size can force a region to contain a nonzero lattice point.
<Minkowski convex body theorem> states that a centrally symmetric convex measurable body $K$ of volume greater than $2^n\operatorname{covol}(\Lambda)$ contains a nonzero point of $\Lambda$. The symmetry and convexity let two lattice-congruent points in $K/2$ produce a difference in $K$.
For $\Lambda=\mathbb Z^2$, a centered disk of radius $R$ has area $\pi R^2$, so the theorem guarantees a nonzero integer point when $R>2/\sqrt\pi$. This sufficient bound need not be the exact first radius where a point occurs: the unit coordinate vectors already have length one. Such estimates connect shortest-vector problems with ideal classes, Diophantine approximation and arithmetic volume bounds.
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